Actually, not really. It's all HTML 5 video elements on the page, so this is all you need:

    .mirrored { transform: scaleX(-1); }

And you can make a button to add/remove that class as needed.

However, I do hope the implementations are consistent as to which way it flips the video.

Changing it is easy. Adding options is an easy way to avoid choices.

But Jabbles' point still remains: It's a hard choice to make.

I don't think it's a hard choice for the browser to make- send the data as it is captured by the camera. How that is displayed is something else, but as timdorr demonstrates, it's not difficult to toggle.

It's a hard choice for the human who has to pick one or the other and justify that decision.

In what way? The developer should have no problem justifying the decision. "It's what the camera sensor sees".

I absolutely understand the application in webchat and why you would want it to be mirrored at a page level, but I'm at a loss to understand why the browser implementation would try to reflect that.

"reflect" tsk boom!

I agree that it's a hard choice; I had to make this exact choice last week when working on a video chat service. Possibly because of the nature of the service (tutoring/teaching) it felt incredibly unnatural to see a non-mirrored image of myself. I tried it out both ways and it really did feel quite peculiar, and I had a hard time figuring out my own movements (kinda like trying to shave using two mirrors).

The last time I tried this it also flipped the controls. Is this still the case?

Are there any usability studies on video self-views? What are the precedents from other video conferencing software like Apple's FaceTime and Google Hangouts?

I expect it heavily depends on the application, context and user role. So it's a great thing to offer developers the choice of whether to mirror or not.

Why would you make it a mirror image? If they have anything written out it would be backwards.

Because people are used to mirrors. Even people who work around video cameras for a living mix up left and right on daily basis.

You see a mirrored image of yourself and a non-mirrored image of the other party.

Why are they different?

Because then the thumbnail where you see your own face behaves more like a real-world mirror, which is easier to reason about if you want to adjust your position.