Hi Alyosha, I'd say forwards. You can also use this expression to say that you have completely fallen inn love with someone "I've fallen head over heels for him, i think he's amazing!" I hope this makes sense. Any questions, please let me know :)

I don't think it specifies forward or backwards.

Usually forward, but it depends on how you fall. The expression "head over heels" is usually used only when talking about falling in love. For normal falling, we usually just say "I fell down/over" unless it was really spectacular.