Front drum (was disc) brake only works when going backwards (!) I am restoring my old bike that hasn't been used for quite a while. The rear brake works fine, however, the front brake has an issue: It only brakes when the bicycle is going backwards, not forwards!

I have no idea how that is even possible - but it shows that at least it can't be the handle or the wire. Something broken inside the brake perhaps? 

I attached a picture of the brake for clarity. Any suggestions on how this is possible and if it can be fixed?
There are two possibilities:

1.) The roller brake mechanism and internals are completely seized. Based on the level of rust on the outside of the brake, there is a strong possibility that there is rust inside as well.

Shimano roller brakes do need to be in fairly good condition, and have the correct Shimano grease in them to work well. If the brake had a very thick type of grease injected into it, it may have gummed up the brake completely.

The Shimano Nexave roller brakes, are technically serviceable, but are not warrantied by Shimano if opened up. They are designed as a complete unit; replacement internals are not available, and must be scavenged from other brakes. Most shops will typically not service the internals; know that if a shop is willing to, there are no new parts going into the brake.

Nexave brakes are fairly inexpensive, and any good shop should be able to supply you with a replacement, and install it as well. Despite my normal recommendations to overhaul existing equipment, this is one instance where replacement is the best route.

2.) You are possibly missing a spacer/locknut between the brake drum and the frame.

While this is not your exact brake, the parts are similar. You may be missing item #2.

It looks as though the end of the fork blade may be contacting the drum, and exerting pressure on the drum, preventing it from springing back. Be sure that there is the appropriate locknut holding the brake unit in place, and the over-locknut dimension is 100mm.

This is only likely if the hub has been overhauled, and the brake was put on incorrectly, with the locknut missing.

The reason the brake works while rolling backwards is that the drum is being pushed in the "open" direction, loosening the internals (from whatever is making them stick), and allowing the brake to operate normally.