A lawn mower may perform better in reverse than forward gear for a number of reasons. When a gas powered lawn mower runs in reverse, its blade continues to spin in the same direction as it would if the mower were operated in the forward gear. Blade sharpness issues, therefore, can be discounted in powered mowers.
A non-powered push mower is a lawn mower consisting of two wheels and interlocking blades. A push model that mows grass better when pulled in reverse than pushed forward may need to be sharpened. The forward motion of the wheels is what turns the interlocking blades and, when operated in reverse, the blades turn in reverse. A mower that mows in only one direction may have one dull and one sharp side on its blades. Have your mower professionally sharpened to ensure the issue is resolved.
Powered mowers don't change the blade's orientation when they're pulled or driven in reverse. Any problem with a powered mower, then, is caused by the manner in which grass is knocked down or folded over as the mower passes on top of the lawn. Ensure there are no obstructions on the front of the mower, such as a piece of plastic or a bent piece of the mower body around the axle.
If your lawn mower has adjustable wheels for cutting higher or lower grass, ensure the back wheels are not set significantly higher or lower than the front wheels. If the back wheels are on a higher setting, and the front wheels are set very low, it's possible the mower is crushing the grass before it reaches the blade when operated in the forward direction. Set all four wheels to the same height and test the operation of the mower in both directions to ensure the problem is resolved.
Most modern riding mowers have a number of safety mechanisms in place to prevent harming yourself or others while mowing. Riding mowers typically have a switch that must be flipped to put the unit in to reverse. Ensure this safety mechanism is functioning properly and is not interfering with the mower's operation in either direction.