Are you allowed to move backward on a monopoly board?
No.  moves after rolling dice are clockwise/forward.
from the rules here

Place your token on the corner marked &quot;GO,&quot; throw the dice and move
your token in the direction of the arrow the number of spaces
indicated by the dice. After you have completed your play, the turn
passes to the left. The tokens remain on the spaces occupied and
proceed from that point on the player's next turn.

The direction of the arrow is pointing clockwise; on future turns you &quot;proceed&quot; from that point.  There is no choice as to direction moved under normal Monopoly rules.
The exceptions are is a card tells you to move backwards. Or if you're playing with the speed dice then

If you roll a three-of-a-kind (all of the dice show the same number),
you can move anywhere you want on the board!

But under standard rules you move clockwise only.