Mine has no phase reversal because each channel has 2 inverting stages so the output is in phase with the input.
No, that's not what is meant by phase reversal, in the context of op-amps. The problem with the TL082 is the inputs exchange functions i.e. the -input becomes the +input and +input, then -input, when the common mode range is exceeded. In other words the phase of the output is reversed, when one of the inputs is taken to within 2V or so of the negative rail. This makes the TL082 prone to latch-up, under certain conditions, as negative feedback, becomes positive, the output voltage will keep moving towards one of the supply rails, until it saturates and stays there. It can be a problem when one of the supplies is lost, or at turn-on as AC coupling capacitors charge.
Magic is right. The NE5532 is a better choice.
I still think you should try transformers. No gain is required, just a higher voltage. If the amplifier is designed to drive low impedance, 8 to 32 Ohm headphones, then it will have no problem driving your 600 Ohm headphones, via a suitable transformer.