While I generally write stories (especially shorter work) more or less in a vague order - I'll start with the earliest scenes, than the middle, then the climax and ending - but apart from that I have a pretty random, haphazard way of writing. I have no qualms about jumping around and moving onto something later in the story if I'm stuck with the scene I'm currently on, and giving yourself the freedom to that is a great way to combat writer's block (at least for me).
Working on later scenes can trigger bursts of inspiration of just how to go about connecting that to earlier parts you've written. And much of the time, particularly in the middle portion of a story, I really have no idea of the order its meant to be in: I often find it effective to just write as the ideas come and work out the strict order of events later (I tend to mess with chronology a lot too, so even if I've worked out the entire plot, the actual order that the plot elements are told in will often move around a lot while revising).
When writing in a somewhat random order, I just keep the word document exactly the same as I would if writing in order: when writing something that comes earlier than stuff I've already written, I just start writing it wherever its supposed to go, and when writing something that comes much later I just put a little placeholder thing in brackets like (other stuff happens here) or whatever and then start writing after that.
I can't say I've ever gone so far to write a story backwards, but it would be an interesting experiment to try. Its probably not a particularly intuitive way to write, but it could be a good way of going about something where, say, you have a good idea for how it ends but are very murky on the events leading up to it.
In regards to continuity, I honestly don't think it would be any more of an issue when writing in a completely random order than writing in direct linear order. No matter what, there'll be a lot of work to be done during the editing process to ensure consistency in continuity, plotting, characterization, and the like. Continuity isn't really a big issue when writing the first draft, its when revising that you need to go through with a fine-toothed comb to iron out all the inconsistencies.