EDIT: From the discussion in the comments, I gather that you're asking about the division of parts of the invisible spectrum into the equivalent of "colors." We have effectively done that in some sense. Different parts of the invisible spectrum have different names. The lowest-frequency ("reddest") waves are radio waves, then microwaves, then infrared rays, then visible light, then ultraviolet, then X-rays, and the highest-frequency ("bluest") waves are called gamma rays. Each of these has a name because it interacts differently with its environment (giving it a different "feel"). Gamma rays are very destructive, ripping apart atoms, while radio waves merely gently move them up and down. (There's actually a part of the electromagnetic spectrum that you can directly feel, even though you can't see it - infrared rays in a certain range of frequencies interact with the water in your body, and you feel heat as a result.)
If you want to divide the various named parts of the spectrum into finer categories, there also exist various naming schemes, and no one scheme is objectively correct. For example, UV rays can be divided into UVA, UVB, and UVC (from lowest to highest frequencies). X-rays can be "soft" or "hard" (where "hard" is higher frequency). Infrared rays can be "near-infrared" (high-frequency) or "far-infrared" (low-frequency). There's a specific category of microwaves called "millimeter-waves" which have wavelengths of about a millimeter. You could also identify "colors" in the different frequency bands based on the filters applied to various types of telescopes, as seen below. For an even more well-defined notion of color, you could consider individual atomic transitions (such as the transition of the electron in a hydrogen atom from its first excited state to its ground state), which have a well-defined frequency, and thus a very specific "color." These colors are typically named according to the characteristics of the transition that generated them (for example, the one I just referred to is called H$\alpha$, or the Lyman-alpha line). For a database of atomic transition lines, look here: https://www.nist.gov/pml/atomic-spectra-database. The original answer below is all about how you get machines (typically telescopes) to perceive these colors, and how to turn the data they perceive into a human-readable form (i.e. a color image).
The act of mapping the electromagnetic spectrum outside the visible range basically defines most of the field of astronomy. Astronomers use infrared, UV, radio, X-ray, gamma ray, and microwave telescopes to do exactly what you're talking about - map the sky throughout the entire invisible range. If you were to turn these telescopes onto ordinary, non-astronomical objects, they would work just as well (this is, in fact, how a lot of telescopes are calibrated).
In order to isolate a particular part of the invisible spectrum, astronomers use lenses and mirrors of different size, shape, and composition. As such, the sensitivity of a device to the invisible spectrum depends heavily on its design. For a UV or infrared telescope, the mirrors will look quite similar to the usual visible-range ones. For radio telescopes, you don't have to have the same tolerances, due to radio waves' much longer wavelength, so their "mirrors" are giant metal paraboloids. Meanwhile, for X-ray and gamma-ray telescopes, since the wavelength of the radiation is about the same size as (or smaller than) the spacing between the atoms in the mirror, reflection and focusing is very difficult, so "mirrors" are usually dense plates placed so that X-rays and gamma-rays will hit them at glancing angles. In addition, the design of the "camera" influences the telescope's response to various parts of the spectrum. For UV and infrared telescopes, a CCD is used, much like in visible-range cameras. In radio telescopes, a radio antenna (or an array of such antennae) is used. In X-ray and gamma-ray telescopes, a scintillating crystal or silicon strip array is used, which both take advantage of the fact that X-rays and gamma-rays are ionizing radiation which would destroy an ordinary CCD.
At this point, you have a device that is sensitive to a certain part of the invisible spectrum. Its output is a black-and-white image representing the intensity of the radiation coming from a particular point. In order to turn that black-and-white image into a color image, astronomers use the same thing that a normal camera CCD uses: filters. Filters further restrict the sensitivity of the device to different parts of the spectrum. They come in both broadband and narrow-band varieties. The broadband filters let in a wide swath of the telescope's sensitive range, corresponding to the "bluer" or "redder" parts of that section of the electromagnetic spectrum. So, to get a reasonably accurate mapping of the invisible spectrum to a color image, you would take three broadband filters in your telescope's sensitivity range. The one sensitive to the longest wavelengths would correspond to the color red; the one sensitive to the shortest wavelengths would correspond to the color blue; and the middle one corresponds to green. Taking black-and-white images with each of these filters, coloring them their respective colors, and layering them on top of each other allows you to map the invisible spectrum onto the visible spectrum.
The narrow-band filters are tuned to only accommodate a very narrow range of wavelengths. These wavelengths correspond to the atomic or molecular transitions of important atoms and molecules in astrophysics, such as neutral hydrogen, carbon monoxide, sodium, or oxygen. Most of the narrow-band filters are in the UV-IR range, since most atomic and molecular transitions are in that range; the only major exception that I know of is the filter that isolates the hyperfine transition in cold diffuse neutral hydrogen, which is in the microwave band with a wavelength of 21cm. Usually these filters are used to highlight specific features, such as the star-forming regions of a galaxy or nebula. When narrow-band filters are used, the colors are assigned somewhat arbitrarily, since they don't correspond in any real sense to the broadband RGB filters in our eyes and in cameras. Many of the most striking astronomical images you'll see are composites of three narrow-band filters.
In summary: it is possible to make a color image of the invisible spectrum by using a device that is sensitive to that spectrum and making a composite of three filters applied to that device.