Cessna 172 pilot receiving instruction reported an NMAC with another aircraft during a training maneuver.

Narrative

This was a training flight on commercial maneuvers with me and a CFI on board. The event began after us finishing a lazy eight and started setting up for a chandelle while turning towards an area of flatter terrain for the subsequent steep spirals we were planning to do. Both of us noticed the traffic flying about two hundred feet lower than us; slightly behind to the right side; parallel to our direction and is climbing. The G1000 sounded an audio alert about the traffic and both of us looked outside the windows for an attempt to gain visual on it. The CFI suggested me to perform a left 360 degree turn to let the traffic pass by; which I agreed and initiated the turn only slightly faster than standard rate (about a 20 degrees bank angle); giving us more of a buffer for separation. Upon finishing 270 degrees of the turn I scanned the instruments; including the map display to verify the position of the traffic; and discovered that said traffic has turned 180 degrees and coming directly at us from the left; only one hundred feet below us and climbing. I immediately initiated a full throttle Vx climb; reaching a +25 degrees pitch up attitude briefly before lowering the AOA to prevent stalling; while looking out the left side window for that traffic. I gained visuals almost immediately and confirmed that we were on a collision course and separation has not been established. I kept the climb until the traffic passed directly under us and upon leveling off; the MFD indicated that traffic was 400 feet below; about which the altitude we gained in the evasive climb. Both me and my CFI made expletive remarks to each other during the climb; and we agreed that the distance was way too close for anyone's comfort. This could easily be catastrophic. Suggestions: should have did a climbing/descending 360 instead of level; as separation on two axis is definitely better than one.