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HF radio (High Frequency; the portion of the radio spectrum extending from 3 to 30 megahertz (MHz),¹ although only some of this slice is available for aeronautical use) is the backbone of long-range aircraft communications, as HF transmissions are reflected off the Earth’s ionosphere, allowing aircraft to communicate with ATC stations well below the horizon. In contrast, VHF (Very High Frequency; 30 to 300 MHz, although, again, most of this is unuseable for aircraft transmissions) and UHF (Ultra High Frequency; 300 MHz to 3 gigahertz [GHz], although, once again, only a small fraction is available for aircraft to use), the backbones of short-range communications with and between civilian (VHF) and military (UHF) aircraft, pass straight through the ionosphere and out into space, making them line-of-sight-only² (and, thus, useless for contacting aircraft located, say, out over the middle of the ocean).⁴

In the context of aircraft communications, another difference between HF (on the one hand) and VHF and UHF (on the other) rears its head. Unlike VHF and UHF, where the space allocated for aircraft communications is in one neat, contiguous bunch, commonly known as the airband (117.975-137 MHz for VHF, 225-399 MHz for UHF⁵), the aircraft-communications portion of the HF range is split up into lots of separate bands, scattered throughout the majority of the HF spectrum; I count thirteen non-contiguous chunks of HF spectrum allocated to “aeronautical mobile” use, at least according to this PDF.⁶ With VHF and UHF, one can simply build a radio whose tuning range is restricted to the VHF or UHF airband (much like how consumer radio receivers have tuning ranges restricted to the AM or FM broadcast bands⁷), but an aeronautical HF radio is, by necessity, going to have a tuning range that covers a huge amount of spectrum allocated to non-aeronautical use (unless you want to only be able to communicate on frequencies lying in one specific HF band out of the thirteen available, which is, for obvious reasons, a poor solution), posing a great risk, if accidentally tuned to the wrong frequency, of causing interference with non-aeronautical HF users.⁸

How do aircraft and air traffic controllers communicating on the HF bands avoid accidentally stepping on other HF users in this manner?

¹: Although, in aeronautical use, the HF range extends down to 2.85 MHz, into what is technically the uppermost portion of the MF (Medium Frequency; 300 kilohertz [KHz] to 3 MHz) range.

²: Technically, VHF and UHF transmissions can be picked up somewhat below the horizon (out to approximately one-third-again line-of-sight range), as radio waves in these frequency bands are refracted by the earth’s atmosphere to some degree, but the point remains that these waves don’t reflect off the ionosphere.³

³: Okay, sometimes they do, but only under extremely unusual ionospheric conditions.

⁴: This isn’t as rigidly true nowadays as it used to be, as the advent of satellite communications has started to allow aircraft to communicate over very long distances with each other and with ATC at previously-ordinarily-nominally-line-of-sight frequencies, but HF radio still remains quite important.

⁵: ...which results in the lower three-sevenths of the UHF airband actually being in the upper VHF range!

⁶: I am aware that this PDF is a few years old; however, given the degree of extreme conservatism where anything that could possibly affect air travel is concerned, the bands of spectrum allocated to aircraft use probably haven’t changed much since then.

⁷: MF and VHF, respectively, in case you were wondering.

⁸: The purposes to which the HF spectrum between the various “aeronautical mobile” bands is allocated include radar, amateur radio (including a few marked “amateur satellite” - apparently not all of the HF spectrum bounces off the ionosphere rather than passing through), ship-to-shore and ship-to-ship communications, several frequency and time signals, the numerous shortwave broadcast bands (nine lying in the portion of the HF spectrum that concerns us here, plus one more located beyond the upper end of the highest-frequency aeronautical HF band), and even one radioastronomy band, in addition to the dozens of bands whose allocations are labelled in the aforementioned PDF as simply “fixed” or “mobile”, without further elaboration (except for the fact that a number of the latter specifically exclude “aeronautical mobile” usage).

We already have a really nice and concise answer to the actual posted question, but the long exposition in the question text shows a few points about use of MF/HF radio in general which are a bit unclear, so let me begin with this wall of text:

Term HF itself:

Although we have the traditional separation of radio frequencies into wavelength-based bands, that distinction isn't actually very useful in real life. This resulted in unofficial extensions to some bands.
Therefore, the "upper" part of the medium frequency band, with upper meaning above the medium frequency broadcast band, got absorbed by the high frequency band. It's quite common for commercial HF radios to start transmitting somewhere between 1.5 MHz and 2 MHz and continue up to above 25 MHz. In German language, for example, there's a term Grenzwelle (meaning border-waves), which represents the region between 1.6 MHz and 4 MHz, since it tends to experience similar propagation characteristics.

Discontinuous HF air bands:

There seems to be a bit of surprise about the discontinuities in the HF air bands, but that's not a bug, it's a feature.

The actual HF frequency planning is very complicated. The propagation on the HF depends on the time of the day, on the time of the year, on the location, which all which affects the amount of exposure to the Sun a region receives. And then, there's the local space weather which affects the HF propagation, and the long-term solar cycles which affect the space weather, and therefore affect the propagation of radio waves on the high frequencies. It's easy to say that the HF radio reflects of the ionosphere, but it's difficult to say at which altitude, and to get the actual frequency assignments from that.

So to stop writing very generally and to get a bit more specific:

Lower frequencies work good at night, higher work better during the day.

Lower frequencies work better during the winter, higher work better during the summer.

Lower frequencies often allow shorter ranges, which higher frequencies allow higher ranges.

The handwaving in the above statements is intentional. Some interesting tools to take a look are for example the Hourly Area Prediction maps based on ionosonde measurements. Australian Space Weather Services have them available here.

The charts mentioned above show which frequencies are suitable for communication depending on the distance. Let's take a look at a chart made at the time I'm writing this answer:

Let's say I'm in Belgrade, and I want to talk to someone in Portugal. I'd need a frequency assignment in at around 12 MHz to bridge the distance. On the other hand, if I wanted to talk to France, I'd be looking at 8 MHz to 10 MHz, and if I needed to communicate locally, I'd be looking at somewhere around 4 MHz to 6 MHz.

Such planning is quite interesting in case you want to use a traditional commercial HF station, such as Stockholm Radio for example. They'll have a set of guard frequencies they listen to, usually one on every band they cover, and a set of operational frequencies they'll use to reduce load. Here's an interesting article (STORadio Pilot Refresh) from them about HF radio.

Contiguous VHF air band:

Let's take a look at sizes of dipole antennas (the point can be extrapolated for other, more common types as well)) in free space for the VHF air band. One of the many calculators says that at 118 MHz, we need a dipole antenna which is around 120 cm long. On the other hand, at 137 MHz it's around 105 cm long. That's a difference, but nothing that can't be solved. This makes it very attractive to have a contiguous band on the VHF.

On the other hand, on the HF, we can't have a contiguous band, due to propagation. If we take a look a the antenna sizes for the highest and lowest guard frequency for the abovementioned STORadio, we have 40.826 m for the 3494 kHz, on one end of the spectrum, and 6.146 m for 23120 kHz the other end of the spectrum. In practice, mobile HF antennas, be they on a ground vehicle, ship, or an aircraft, are often made with trade-offs in mind, trying to work okayish on one of the more important bands, and having a so-called antenna tuner to match the impedance mismatch, enabling the antenna to work on other frequencies. The reduction in antenna efficiency is usually off-set with large enough transmit power.

So to sum this section up:

We don't want to have contiguous HF frequencies, due to HF propagation, and due to different frequencies, we have to make trade-offs on the radio set-up. On VHF, we have technical reasons which make contiguous frequencies attractive.

Finally, I'll rant a bit about the culture on the HF:

First of all, the statement made in the question claiming that the radio has to either continuously cover all HF bands, or work on only one is incorrect.

Let's take a look a the maritime mobile radio service. It also has VHF and MF/HF sections. The VHF radios are channelized, just like air band radios, but instead of showing the frequency, and channel step, they show the channel name. The HF radios behave the same way: There are the so-called ITU channels. For example, the channel 1201 is the first channel in the 12 MHz marine band, and has a set frequency for the ship to transmit, on which the coastal station listens, and a set frequency on which the ship listens and the coastal station transmits. Depending on the radio, you get a channel knob, which you use to change the channels, or you can use the menu to select the band, and a channel within the band.

For amateur radio, it's similar: There's a band-selection menu, or a set of buttons, which allow you to select a band, and then you can chose the frequencies within it to transmit.

On land-mobile HF radios, the situation is a bit different, since there's usually a set of pre-programmed frequencies to which the radio can tune. Quite often, they'll even have methods to automatically chose the best band from the list of available frequencies.

However, what's common to all 3 is that, usually there's a more or less difficult way to "unlock" the radio. The radios usually cover almost entire HF range, and are limited "only" by the software to the licensed frequencies.

Finally, I'd like to address the great risk statement, which is also, in my opinion a part of the culture.

Namely, it's important to understand, that when using the ionospheric refraction to communicate over HF (as it's intended to be used), HF radio is not reliable! There's an effect called fading, which leads to differences in signal strength. It can vary with time, and with frequency. Sometimes, it's very slow, compared to the rate of change of the signal, and sometimes, it can be very fast. Here's an example of fading on an AM broadcast station in the 31 m HF band:

The top part of the picture, right below the 9500 (frequency in kHz), shows the current spectrum, with each grid being 2 kHz wide and 10 dB of signal strength tall. The bottom part with the red line shows the so-called waterfall, view as if you were looking at the spectrum from above, showing around last 5 or so seconds. Yellow is the audio signal of the station, with blue stripes representing the fading, which was around 20 dB in intensity, a reduction of signal power of around 100 times, on those frequencies, which is enough to reduce a good quality audio to unreadable level.

There's interference from other HF users, which can be intentional and unintentional.

Here's an example of a typical day (well, night actually) on an HF broadcast band:


We can see several broadcast stations, running AM, we can see a regular radar radiating right next to a couple of very weak AM stations, and we can see the usual behavior of ionosondes, covering very wide spectrum very quickly, with very little regard to other users. It's actually sweeping continuously through frequencies, the snapshot dots making a line are artifacts from the radio.

Then, there are other intruders from out of band: It's not uncommon to see stations operating outside of their allocated frequency bands intentionally. It's not uncommon to see encrypted military HF communications in amateur radio band and radars in amateur bands, and it's also not uncommon to see pirate broadcasters in marine bands as well, transmitting either music, or chatting, or transmitting propaganda. The situation is so bad, that the ITU even has "Regular Monitoring Program in frequency bands between 2 850 kHz and 28 000 kHz" just for that. You can even take a look at their reports. It's quite common to see descriptions such as GERMAN SW PIRATE, or TWO MALES IN USB causing interference to digital selective calling frequencies, or INTRUDERS LSB and so on.

So to sum up the risk part: On HF, you'll never know if you're going to be struck by a radar, by a numbers station, by a pirate, or by regular fading. Therefore, you need to take steps to recover from such situations and to be able to continue on. This leads me to conclude, that the risk isn't great at all. If you actually transmit for a bit outside of your own assigned band, nothing is going to happen. Yes, you might jam someone for a bit, but fading could have made the same effect as well. The jammed party is just going to have to say the last relation again and that's more or less that.

By the way, there is a trend of replacing HF communications with satellite communications, which seem to suffer much less problems compared to HF.

Oceanic airspace is divided into MWARA, Major World Air Route Areas. Each of these areas defines a set of HF frequencies that are used for air-ground communications. Within each area, the frequency used by aircraft at any given time is usually predetermined by the time of day based on propagation characteristics of the given frequencies (and published in a schedule).
For example, this page lists hours of operation for some areas: North Atlantic (NAT) MWARA Profile and Frequencies.

Typically a ground station will monitor all of their MWARA frequencies simultaneously. (This is at least true for the South Pacific area, which I'm familiar with.) So if an aircraft tries to reach a ground station and cannot, then they might try a different frequency, or just try later. Frequencies other than the assigned MWARA ones are not monitored.

The ground stations use equipment that cannot tune to a frequency other than the assigned ones (e.g. five in the case of the SP area). I'm less familiar with the HF equipment installed in the aircraft, but since there is a fixed list of published frequencies worldwide (a few dozen), the frequency may be selectable from a predefined list.