If you want to get to the root of the problem then probably best to follow the suggestions for anti-skate and cartridge setup first. If you don't mind the reversal of left / right channels then that's fine and will result in no other undesirable artefacts.
The bit I don't get is that swapping L/R channels results in a better balance, you'd think it would just be out of balance in reverse.
Final thing... is it just your TT that sounds out of balance? Left/Right tracking on potentiometers is pretty poor and changes with the volume setting so if other sources are out of balance and you have a potentiometer for a volume control it might be worth checking that as well.
The bit I don't get is that swapping L/R channels results in a better balance, you'd think it would just be out of balance in reverse.
Are you aware of phasing, absolute polarity and channel orientation?In order to change the phase you'd need to reverse the connections at one end of the RCA connector... I don't think anyone suggested that. Polarity is just phase referenced around 180˚ steps, so that won't be a problem either. Channel orientation will of course be affected (unless the outputs of the pre are swapped as per @russ69's suggestion).
Final thing... is it just your TT that sounds out of balance? Left/Right tracking on potentiometers is pretty poor and changes with the volume setting so if other sources are out of balance and you have a potentiometer for a volume control it might be worth checking that as well.