Look at the linked demonstration diagram very carefully.
Notice that both switches are shown mounted upright.
Notice that the two traveler wires do not connect to the same terminals on both switches while being installed.

Here is a link that might be useful: Youtube examples

From my understanding of a three way switch it should work with the travelers on either the same or opposite terminals. It will just change whether the light turns on from the switches being in the same or opposite position.

Yes, you are correct, reversing the travelers will just change the switch direction.

Since there is obviously a problem here - you can not make assumptions.

You just need to start at Switch A and disconnect the two travelers. Using an an AC voltmeter referencing neutral or ground, measure the common again that you believe is the line side - still got voltage?

Next, measure the two switch terminals - one should be hot and the other not. Switch the switch other way and it should alternate. (If not, Switch A is faulty or miswired.)

Next measure the two bare travelers, since they are going to Switch B and then on to the load, there should be no voltage. (If one of them has voltage, there is a wiring issue here and you need to confirm that they are really going to Switch B and then on to the load.)

Reconnect the travelers on Switch A and disconnect the travelers on Switch B. Toggle Switch A and measure the travelers at the Switch B box - should see them alternate hot.

Reconnect Switch B and you should see it's common terminal, switch on and off when you toggle Switch B. (If not Switch B is faulty or miswired.)

BTW - I am assuming you know when to de-energize and re-energize the circuit via the circuit breaker. :)

Thanks for the help. The switch was bad. I replaced it and everything is working properly.

That's great!
Good for you!