Do restriction enzymes cut both forward and backward?
Neither I think
The question implies more or less that RE's (Restriction Enzymes) run down the double Helix much like a train running down the track.
This appears true for some enzymes like RNA- and DNA Polymerases and e.g. Reverse Transcriptase, but it is not how RE's work.
You can do a partial digest of a genome, which will result in a few bands. If the incubation time is really short you will only get a few large fragments. The longer the incubation time is, the fragments will get smaller as the digestion nears completion.
If you run an Agarose Gel after a short digestion time you will see that the pattern will be made up of every conceivable combination of bands, ranging from the complete genome ( no cut in the case of circular, e.g. pBR322), via 1 cut (once again the complete genome but linear) to a range of n-cut fragments.
The appearance of every conceivable combination of cuts indicates that the cutting is random.
Therefore, the RE won't run "down the tracks" but will handle the Double Helix like any other Enzyme treats a substrate: Attach, Perform, Release....
Due to Steric Hindrance (e.g. DNA Supercoiling) not all sites are equally accessible to any given RE. Moreover, there are different Classes of RE's. So don't be surprised if some bands in a partial digest show up less convincingly than others....