Catadioptric lenses are lenses that combine both refractive and mirror elements. A simple mirror lens with no refractive elements is not a catadioptric lens. I've never seen a commercially available photographic lens with only simple mirrors, though.

Simple front coated mirror lenses, such as those used in reflector telescopes, that are clean do not demonstrate flare. Off axis light is, by definition, reflected outside the lens' field of view. Light from within the FoV is only reflected to a single point (or a blur circle for out of focus light sources) at the lens' focal length. The reason many simple reflector telescopes have barrels is to protect the mirror from condensation, dust, and damage. Dust and condensation on the surface of a mirror lens have optical properties than can, in the presence of off-axis light, reduce the contrast of the image projected by the mirror in a way that resembles veiling flare, but that is due to the nature of the substances on the mirror, not to the nature of the mirror itself. If the barrel of such a telescope is reflective, the reflections from the inside of the barrel can fall on the mirror and cause flare.

Any flare created by a catadioptric lens is caused by its refractive elements. This could include corrector plates, the glass in front of a back coated mirror, or even flat protective plates at the very front of the lens. Because most catadioptric lenses used for photography have fewer refractive elements than typical refractive lenses created for creative photographic use, most catadioptric lenses demonstrate less flare than many refractive lenses used in creative photography. But what flare they demonstrate is caused by the same properties of refractive lenses as the flare produced by a non-mirror lens. Thus any flare demonstrated by a catadioptric lens would be similar in nature to flare demonstrated by a refractive lens.

Catadioptric lenses with fewer refractive surfaces are more resistant to flare. In shooting situations that would induce ghosting with many conventional lenses, a mirror lens does not show the same tendency:

Normally we would expect to see some ghosting in the area, indicated by the yellow oval, directly across the center point in the photo from the brightest light source in the scene. Yet there's not even a hint of ghosting in this image taken with a Nikkor 500mm f/8 Reflex.

If you overexpose the scene enough, you'll get ghosting that looks exactly the same as ghosting with a typical refractive lens, other than the higher difference in brightness between the primary image and the ghost refelection:

The moon is well beyond totally blown out before we see a ghost reflection against the dark empty sky. Image taken using an Opteka 500mm f/6.3 + 2X teleconverter.

This image was taken with a Samyang 500mm f/8 mirror directly into the setting sun with enough exposure to see some flare. It's fairly conventional looking flare.

The veiling flare in this image from a 500mm f/8 mirror is very well controlled but otherwise the same as what one would see with a more conventional telephoto lens. The bokeh, however, is not!

We see the same thing in this one from a Minolta RF Rokkor 250mm f/5.6. Other than the degree to which the flare is controlled, there is no difference from a non-mirror lens under similar conditions.

(source: staticflickr.com)

Taken with a Walimex 300mm f/6.3 mirror, this sunset shows no obvious flare. A regular refractive lens of this focal length placed in the same situation almost certainly would.