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I had two Lutron 3-way dimmer switches for my bedroom lights. They looked something like this: https://www.amazon.com/Lutron-MAW603RH-WH-Electronics-Maestro-Dimmer/dp/B000BQMVXC

I replaced the bulbs w/Philips Hue smart lightbulbs, so I needed non-dimmer switches. I bought two simple 3-way rocker switches, and tried to wire them the same way. Unfortunately, on my first try, something was wrong. When the 1st switch was up, 2nd switch could turn the lights on and off. But when the 1st switch was down, lights were off and 2nd switch couldn't turn the lights back on.

So I tried moving wires around on the 2nd switch, and they behave as expected -- I can turn on/off the lights from both switches.

Does the mere fact that switches behave as expected show that I wired these two switches correctly? I ask because I'm a new homeowner, and am just paranoid that I may have mis-wired something that will burn down the house one day. Thank you!

(Ground wires are easy to identify, so you can assume I wired them correctly)

(There were no neutral wires on both my old and new switches)

You now have them correct, the one good thing with 3 wires connecting the 3 way switches you don't have to worry about burning things down they just don't work as you found out if they are wrong. Good job of figuring it out, you would not believe how many times I have been called to do what you just figured out.

Yes, it's fairly easy to confuse the common terminal with the traveler terminals, becuase they are in a different position on every model of switch.

The rule of thumb is that the 2 travelers always go into the same cable, whereas the common will not, at least not at one end.

So for instance if one end all 3 wires (red white black) go into the same cable, and at the other end red and black go to the same cable and the other black comes from a different cable, red and black are your travelers.

I am a huge fan of marking travelers with yellow (or whatever) tape. There is no need to distinguish them from each other, but two wires tagged same-color in the same cable are a dead giveaway that they are travelers.

It sounds like you mixed up a common terminal with a traveler terminal

The misbehavior of your original, incorrect installation makes me think that you landed your common on a traveler terminal at one end (the end that was stopping the other switch from working), while one of the traveler wires was landed on the common terminal (on a 3 way switch, this is always the differently colored non-ground screw, don't pay attention to where on the switch it is).

As a result, that miswired switch stopped power from getting to the lights entirely when flipped to one position, as the common wire on that end was disconnected, while the two travellers were connected to each other and nothing else. When you flipped the miswired switch to the other position, it then connected the common to the miswired traveler, allowing the other switch to control the light normally. Now that it's fixed, there's no need to worry about the house burning down as a result of this particular oops.