Probably. What brand and model is the valve?
Another potential cross-over is for the washing machine supply lines. Not normally an issue, but possibly depending on what you've done there. If you have shutoff valves to the WM, turn them off and see if it makes any difference. As was said early on, a hot water recirculation system can have that problem. If your piping is ideally routed, it can be done without a pump, but you'd be able to see some return line connected down low to the water heater, and it doesn't appear there is a line there, so probably not in your case.
If you have a two-handle valve with a hose connection on the outlet that has an individual shutoff, that can cause it as well. ANy place where the hot and cold are connected together is suspect. Even a two-handle shower valve if you have a shutoff at the showerhead and leave both valves normally open.
I have a hose that is connected to both hot and cold, and at the end of the hose is a typical nozzle that can be turned off. I use it to get warm/hot water to wash the car. While that's being used, there is the potential of the hot and cold mixing in the rest of the house, most noticeable when the nozzle is closed. I always shut the valves off when finished, so the cross-over is blocked.