Repeatable? Definitely.

Using the same film, cross-processing has perfectly repeatable results, but it helps to be specific: the negatives will turn out the same, positives (virtually always scans) may not, and it's the positives you see on Flickr and elsewhere.

The reasons for this are pretty simple: both the film emulsions and processing chemistry (both E-6 and C-41) are highly standardised. If either one changed unexpectedly to any degree, you'd get unpredictable results regardless of whether you were cross-processing.

Scans (especially of E-6 in C-41, but the reverse as well to some degree) admit much more variation - if done at the lab, it depends on the machine, it's presets, how much effort the operator takes, and whether they're familiar with cross-processed negatives. When people scan at home, they might do so with different goals in mind - some may want the extreme colour casts, some may prefer to correct the image to almost "normal", and anywhere in between. The results are like you see: the same film (even the same image) can have wildly varying results depending on its treatment.

I should add that this is true to some degree of any negative film – there is always much more room for variation in the colour balance and treatment when converting to positive than when shooting direct positive film.

Why? Good question.

I don't think there's a general answer to "why" - often it's just a matter of enjoying the look (like with any other film/technique), following a fad (they've happened a few times), or even not having easy access to E-6 processing. If there is a general reason "why," then I don't think it's significantly different to any other film. For example, original Agfa Precisa1 yielded a high-contrast image with exaggerated blues, but otherwise mostly neutral colour shifts, and an almost-clear base. As a result, it was reputedly used for commercial shots of blue jeans.

Examples

The above makes "realistic" examples difficult, but here's some mine (otherwise it's impossible to comment on processing/lighting) to display the kind of range you can get, shot on Precisa:

(from left to right)

• Very cool lighting, a window on the shaded side of a building, almost "straight" and very typical Precisa blue.
• "Golden hour" lighting; very warm, corrected (maybe even over-corrected?)
• Fluorescent lighting at night; corrected, but mostly for the green of the lights, not the cast of the film.
• For comparison, non-cross-processed Precisa in cloudy/cool light.

Going back to the question of repeatability, even with such a broad range of outcomes, in the same situation I'd have every confidence of getting the same result.

1 Current new/in-date Agfa Precisa is not manufactured by Agfa, but is a different emulsion packaged under the brand. It's reported to have a greenish cast when cross-processed (I haven't tried it myself).