Unlike IR LEDs which can be invisible or seen as a very dim red depending of the wavelength, UV LED can be seen even at wavelengths like 365nm, which is invisible to the human eyes.
395nm LEDs looks like a strong bluish white color, and 365nm LEDs looks like a dim white color which changed to greenish color when lowering the current. 395nm LED beam looks strong violet, while 365nm LED beam looks dim white color.
And the spectrum of both have continuous radiation from the blue to the red.
Why this is happening?
@James, @RRK or @Medved can be know?

There are three proncipal ways to cause the observation:
1) Are you sure it is the real spectrum of the LED?
The thing is, it could just well be an artifact of the used equipment, mainly when it relies on an interference effects to filter/determine the colors:
E.g. a 365nm radiation line passes to the same places as a 730nm line would, so unless there are explicite measures in the equipment (a filter, correlation with where a notch needs tobe,...) to suppress these effects, that 365nm LED UV radiation makes the sameimprint as the730nm red would do.
It is thesame effects that makes ID LEDradiationto look pink at cameras - beside the red channel, the long wavelength passes a subpixel filter designed to select the blue, as it has double wavelength...

2) Other cause could be many materials to actually act as kind of phosphors, although extremely inefficient to be really useful that way, still generating enough to become visible.

3) Some impurities in the LED semicoductor junction may cause energetic traps, allowing more energy transitions, so generating also other wavelengths than the thing is primarily designed for. Again, splitting the energy to two similar energy states means photons would fall around the red part.

@Medved: Here is a spectrum of my 365nm UVA LED flashlight.
You can see that there is a continuous radiation from the red to the violent.

And if you look at e.g. blue or near-UV (purple,...) LED there is no such "continuum" visible along the main line?
Or when looking to different source with similar porincipal UV line (e.g. BLB,...), is the same "continuum" still present?

The question still is, whether, whether it is real, or it is parasitic response due to e.g. scattering on the grating (then exciting across the complete sensor)...

If this radiation is real, it should be observable also directly by eyes, e.g. in the reflection from CD or DVD (by eyes you may distinguish whether there is real continuum "rainbow", vs a smeared bluish glow in case of the scattering).
Or you may use some UV (yellow gel) filter to really suppress the UV and see whether some of the yellow remains (it will be yellow in case the continuum is really there)

The thing is, even a small parasitic response of the sensor to the UV outside of its working range may create a "significant" image...

The thing is, what I've seen, no "parasitic continuum radiation" was ever mentioned, on the contrary, the related materials express the main advantage of the LED to really be free of parasitic radiation (unlike green/blue lines on MV discharge,...), therefore I'm a bit skeptical about this being real, other than an artifact/limitation of the measurement setup/equipment.

I see this continuous radiation also with 395nm LEDs, but now with blue LEDs.
That continuous radiation is real. You can see my attachments in my post. One of them is a closeup of the 365nm UVA LED in my UV flashlight. The die glowing dim white although the plastic chassis which the LED is mounted, slightly fluorescence sky blue color. But most of the visible radiation comes from the white glowing of the die itself.
Blacklight fluorescents have only the visible mercury lines in their spectra.
@James may help?

The continuum of visible radiation is normal with many UV LEDs, and can be quite a problem.  There are two main causes : fluorescence of the sapphire substrate used for the chips, and presence of impurities in the p-n junction having lower band gap than the high voltage AlInGaN dopants used for UV radiation.

The same parasitic continuum can also be found in blue and other colour LEDs, but to a much lesser extent.  On the one hand, the impurity content tends to be similar but because the forward voltage of visible LEDs is lower than UV LEDs there is less excitation of the impurity traps.  Secondly, the visible LEDs have efficiencies up to about 85% whereas many UV LEDs don’t even reach 5% efficiency.  So the intended radiation with colour LEDs drowns out the tiny traces of parasitic continuum and we hardly see it, but for UV LEDs it can be a substantial amount of the radiant flux.  Thirdly the parasitic radiation tends to cover the visible wavelengths only - and it overlaps the intended radiation with visible LEDs so is much harder to distinguish.  But for UV we cannot see their peak radiation and only observe the parasitic glow.

This can be a commercial problem because customers expect UV LEDs to be either invisible or a pale blue/violet, like traditional lamps.  However, since the parasitic continuum tends to occur in two broad bands peaking in the green and sometimes also in the red, many UV LEDs have an unsightly white hue that is slightly greenish or yellowish.  Customers hate this.  It’s even more of a problem because the parasitic radiation is caused by tiny variations in impurity content which are different from one LED to the next or between batches.  For instance, in UV LED T8 tubes the individual LED spots can all have slightly different colours.  This looks cheap or defective.

UV LED manufacturers have to make extra efforts to control the impurities, which makes them far more expensive than other LEDs.  Or they resort to colour binning and may scrap as much as 50% of the production, keeping only those within a narrow range of colour appearances. 

I had the same problem when developing UV LED filaments some years ago, and now have a patent on a much cheaper solution.  I add a tiny amount of a blue-emitting phosphor which converts about 1% of the invisible UV to a rich blue radiation.  That completely overpowers the parasitic radiation and delivers a nice-looking blue glow.  We can now use 100% of the UV LED chip yield which greatly reduces costs, and there are no extra costs needed for impurity process control or colour binning.  Recently, another trick is to add an intentional dopant to the p-n junction to generate a small amount of blue light.  The result is an LED chip having a dual wavelength emission.  Due to the patent situation though, not all companies are able to offer consistent colour UV LEDs and there are a lot of rather bad looking products.