C172 flight instructor reported taking evasive action to avoid another aircraft while in the VFR pattern and talking to Tower. A subsequent ATC instruction placed the reporter closer to the other aircraft.

Narrative

After our touch and go; we were told to extend upwind for spacing and our crosswind would be called. Tower called our crosswind over point defiance. The student and I were mid field downwind; right close traffic for Runway XX in ZZZ. The student and I were in the middle of our before landing checklist when we noticed another plane on what looked like a crosswind flying directly at us. We banked left away from traffic and descended; but still continued to fly the pattern. I was not sure if they were closed traffic of just east bound departure. The other traffic continued to fly pattern as well but they were a couple hundred feet above us. I believe Tower became over saturated with aircraft and forgot about us or vice versa. At this point Tower had me perform a left 360 which placed me closer to the other aircraft followed by a right 360 over the approach end of Runway XX. Once spacing was obtained we were both told to land but no number was given to us to contact. This occurrence was not due to pilot error. I spoke with the other pilot and she agrees. Both PIC (Pilot in Command) in each plane were CFI and I believe that helped the situation.