Whilst I can't speak for a T500, on later Suzukis your issue would suggest that there is a problem with the self centering mechanism on the gear shifter. Age, wear & tear or someone standing full weight on the shift lever are the usual reasons for this.
Hopefully it is just a simple spring that has failed, or is way too limp. The selector arm locates over pins in the selector drum cam plate - this has a centralising spring one end of which is retained in the crankcase. Pointless me talking pawl plates, thrust springs etc, this would need investigation by stripping back the LHS of the engine/gearbox. It's a simple enough job on most bikes, but you would need a workshop manual to clarify the procedure for a DIY fix.
Worst case scenario is a bent selector fork or damaged guide pin , but lets not go there for now.