There is a setting in Skype that allows you to project a mirrored image. Find that and change it.
Well, good news and bad news. The good news is that my integrated webcam does that same thing in the Skype app and also in the camera app. I see a mirror image of myself and background. The bad news is that this is built into Windows 8.1 by design, and there is no way I can find to change it.
However, other Skype users report that the other end of your video call will see the image correctly, not mirrored. But let me know if that is not the case.
Geoff, yes, I'd be interested to hear if you can resolve it.
And I saw someone remark about the forward versus rear-facing camera. But honestly, why would it make any difference? Both cameras on my phone project a correct, not mirror, image. No reason PCs shouldn't work the same way.