When I turn on the cold water tap, cold water comes out initially, but quickly and momentarily becomes hot for about 30 seconds, then cold again. At the hot water heater, the cold water feed pipe feels cold to the touch when I turn on the hot water faucet - which makes sense and seems normal. If I turn off the hot water faucet and turn on the cold water faucet, the cold water feed pipe to the hot water heater feels hot to the touch and hot water comes out of the cold water faucet - which doesn't make sense. After 30 seconds, cold water will come out and the cold water feeder pipe to the hot water tank will feel cold to the touch - as it should be. So it seems the flow of water from the hot water tank reverses momentarily, then corrects to the proper flow direction, perhaps when water pressure equalizes. Is that messed up or what? Does my water heater have a broken check-valve or some other type of pressure equalizing/relieving device?

Install back-flow preventer on cold water nipple at your w/h. It seems to me that you have thermal expension problem. Your system pressurizes pushing hot water to the opposite direction. Once the excess of pressure is going to, all returns to normal.

The back flow preventer you want to install is the one for VERTICAL installation. Don't try to install the one for horizontal installation since it will not eliminate your problem.