Changing out an old light switch

Some background on this: the house is old, 1960's, and the previous owners loved having multiple switches run a single light.

I am changing out a light switch inside the house that runs an outside light and this light has two switches, one in the garage and one in the dining room. Yeah very odd spot.

The new switches I see have 3 screws on them, 1 for the hot wire and two for travelers. But the old switch has 2 white wires coming in and 2 black wires. One black wire is hot and one white wire is hot. How on earth do I wire the new switch?

In the pics below I tapped the hot wires.

and second pic