What light switch combination do I need?
This is two three-way switches from the looks of things, but wired in a nonstandard fashion
From the wires you have, this does appear to be a fairly basic three-way switch circuit with both switches on loops from the light fixture, but with the way the second switch is wired, and the way you were able to get it working, it's not a standard three-way switch.  Instead, it's what's called a &quot;California&quot; or &quot;Coast&quot; 3-way setup that brings always-hot to a traveler terminal on both switches and switched-hot to the other traveler terminal on the switches, then connects their commons together.  This yields a working 3-way switch arrangement, as depicted below (illustration courtesy of Wtshymanski/Wikipedia), and has the advantage of providing always-hot and switched-hot at both switch locations, but has the downside that you can't add another switch to it.