If time can go faster or slower without our being conscious of it, can it also run backwards unbeknownst to us?

Yes, absolutely. If it could do that we probably wouldn't know it, since all of our timing devices rely on time to operate and we wouldn't have a reference to compare against.

Time goes slower for a fast moving reference frame relative to another frame. This can be noticed and detected. But time never goes backward relative to any frame. 

What we can barely  see, is our getting older ....

Time only goes faster or slower for us relative to the rest of the universe. That's a very important aspect of this. We still experiece time at the same rate, it just may not be at the same rate as everywhere/everything else. But no, time cannot elapse backwards, that's a very significant aspect of time.

Time is a concept. The "rules" of time don't allow it to go backwards.