Normal 3-way switches are just single pole double throw (SPDT or 1P2T) switches.

In (what I think is) your case, the hot wire comes into one of the switches where it's switched between two connectors. Those outputs are connected to the equivalent connectors on the other side, and then the hot comes out of it and goes to the lamp.

Call the throw connectors for each switch 1 and 2, and assume that the 1s and 2s are connected together. There are four situations: 11, 12, 21, 22. In the case of 11 and 22, the electricity flows from the hot, out the selected connector, where the equivalent connector on the other switch is connected to its hot, and the electricity makes it to the lamp, where the circuit is completed by the neutral that runs "straight" there. (In reality, there are going to be intermediate connections for the neutral.) In the case of 12 and 21, the flow of electricity is going to be interrupted by one of the switches being in the wrong position to transfer the electricity and make the circuit complete.

Of course, none of this really helps solve the problem with your new switches, but it might at least help you understand what's going on.