Sounds like something is amiss. Try using a multimeter and somebody else to activate all the lights whilst you identify which is which. If you know how to use a multimeter it should be straight forward, you are looking for which lights are 12v.

First identify the reverse cable/socket and see where it leads.

Are these red lights the rear fog light? Or are the new LED lights giving you the option to swap a fog light for a reverse light? Some investigation needed I think.

A certain way round it would be to re-wire the reverse lamps using scotch blocks, its fairly easy to do but it looks butchered!