Can backward and forward be used interchangeably with backwards and forwards, or is there some particular situation in which one pair is consistently used over the other?
In British English (and possibly others), backward and forward are adjectives, and backwards/forwards are adverbs.
A person may be backward1 or forward; a car may be forward of a "Stop" line; one might move something forward (that is, to a position which is forward of its current position). However, when describing that movement itself, it's forwards.
Related question: Meaning of "backwards"
1 Using backward to describe some developmental abnormality is frowned upon and not recommended. In fact, describing someone as forward is rather dated, too.