Why won’t the new water valves turn on

This might not be the answer, but for reference here is a problem that can occur if the plumbing is not installed properly.

If you look at Fig. 3, this is what a proper drain looks like. You'll notice the orange line represents the water level in the system, the water levels out in the trap preventing sewer gases from entering the sink drain.

In Fig. 1, you can see what will happen if the drain line has to go up before meeting the main drain line. Again the orange line represents the water level in the system, and as you can see the water level is much higher in the drain.

Fig. 2 shows what could happen if the sink was installed lower than the rise in the drain line. The water would never fully drain from the sink, since it would require extra pressure in the system to push the water up the drain.

I wouldn't guess this is the issue in this case; since as you can see, if you opened the trap in this case you would have more water drain than what is typically in the trap.

In a perfect world Fig.1 and 2 would never pass inspection, but as we all know not all plumbing is properly installed and/or inspected. So this could possibly be the case in this situation (given the plumbing was installed by an amateur plumber), but it is doubtful. It would be more likely caused by a clog in the line.

This is contra turn taps usually used for lever handle taps so they both turn away from the wall on a basin or turn both down in a shower or similar if used in vertical eg. Shower above and below you would get normal taps not contras so that they both turn in the same direction which are the ones that you are used to. Also re reading your post it seems that if you have lever handle taps the hot and cold taps shoul be swapped to make them both turn on when turn towards the centre of the basin.