Question about mounting two LED strips in parallel, one properly and one reversed, and using a switch to switch between the two
better option is to work with 2 positive rails and shared negative, using a switch to choose which positive rail to turn on.
negative from leds goes directly to power supply, positive goes to switch.


a two position switch (on-on)
a three position switch (on-off-on)
connect the power supply in the middle terminal, one positive going to each led rail on side terminals.


but you can do what you are thinking and reverse the voltage, just use a two or three position switch with double terminal and connect like the image (the motor there will be the leds).
about the supply, it's correct what you said, leds consumes no current in reverse.