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 "GO," 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 "proceed" 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.
There is one "Chance" card that says "Go back three spaces." That's the only instance of "backward" movement.
Also, there are Chance and Community Chest cards that say, "Go to jail. Go directly to jail. Do not pass Go. Do not collect $200." These could be interpreted as backward or "diagonal" movements across the center of the board (not around the perimeter).
Otherwise, all movement is forward.