C182 skydive pilot reported a NMAC during initial climb with a non-reporting opposite direction landing aircraft. Pilot executed an evasive maneuver to avoid a collision.

Narrative

After picking up my load of jumpers for the day; I announced over CTAF that I was taxiing from the North hangars to Runway XX. While taxiing; I noted the windsock and the weather broadcasting system indicated a direct crosswind out of 090. I knew the wind had been favoring Runway XX through the day and continued the long taxi to XX instead of taking runway XY. After finishing my before-takeoff-flow; I announced a departure from XX with jumpers. I checked XX's final approach and the runway for traffic. Once we lifted off; I wiggled my butt in the seat; climbed out at Vy; and watched for my landing options should an emergency occur (as I always do in any plane). While on the ground and take-off roll; I did not specifically look for traffic coming into Runway XY. And I cannot see over the aircraft nose during initial climb out. As I climbed through approximately 300-400 ft. AGL movement caught my eye and I saw a Cessna 172 in the right bank under me; turning to the West. I recognized the rental aircraft from the FBO and called the tail number over the radio 'are you on the radio?' No response. I double checked that I was on the CTAF. I was. As I climbed up and around the airport; I heard the 172 call final for XX; watched as they did a touch and go; and then as they turned into right traffic (not standard) for Runway XX. Perplexing behavior. After completing the jump run and starting my descent back to the airport; I called the UNICOM for a radio check. The person in the FBO said I was loud and clear and that they had heard all my radio calls.On the ground; witnesses in the FBO said they did not hear the 172s radio calls; but did hear mine. When I first saw the aircraft pass under me; I thought they had been on a left base. Witnesses on the ground said they had been on final and had turned to avoid me. When I talked to the pilot of the 172; they did not describe how the near miss happened from their perspective. They only discussed the radio communication/lack thereof. They said their headset was new and that they had made 5 and 2 mile radio calls. They also said they entered on a left base; not on a left downwind when entering the pattern. They downplayed the situation and said that 'we weren't that close; maybe 200' implying that it was fine. They said they didn't hear any of my radio calls. I asked if they were on the right frequency. They said they were. They had just come from another airport where they had dealt with a flat tire and they heard the radio calls clearly. In retrospect; having noted the direct crosswind and the possibility for use of either runway; I should have intentionally looked for traffic in the pattern for the other runway. I should always do that; regardless of the winds. I can not know if I heard the complete truth from the 172 pilot. Perhaps they were not using push-to-talk when they thought they were. Perhaps the long day waiting for a tire change had made them frazzled.