Glad to see many others here have far more experience than I on the J2. My introduction was at Gillespie field in SoCal in around 1982. I was towing banners down the beach, and giving aerobatic rides in my Citabria, and was always interested in new, or different aircraft. There was a J2 owner/operator on the field, and try as I might, I wasn't able to cross paths with him until one day I saw him at the fuel depot. I pulled in, and chatted with him for a bit as we fueled, and boldly asked if I could go with him sometime. He said we could go up right now, as he was just doing some landing practice, and extra weight might be a good thing(little did I understand at the time about the top-heavy condition). So I parked my flivver and hopped in his plane. It was loud just rolling around on the taxiway, but when we got to take off it was unbearable. I think the cabin actually acted as a resonator for the engine and prop vibrations. Just deafening.
Well, we took off fine, and went out on our way. I had a Benson B-8M gyrocopter before in which I self-taught to fly. So, I had a basic understanding of gyrocopter flight, but was by no means an expert. He let me fly it for a while, and it was pretty nice despite the horrific noise. Then we joined the circuit for our landing and he rapidly got rather tense, and focused. We had wind directly down the runway at 5Kts. The first landing was a long rolling landing, as he said he liked to taxi back. I found out, that long rollouts were a means to keep full authority on the rotor until stopping without the nose gear touching. It didn't make sense to me at the time, but in retrospect, it was very clear. The landing was similar to what the tailwheel pilot would do, except we had power on and the cyclic back to keep weight on the mains, and keep airflow over the rudder for directional control.
After a few pattern landings, we took off and I flew again for a while. I wanted to see what the 'stall' or more accurately the zero-airspeed conditions were like. Well, we were about 2000AGL over a valley, and I pulled it up some as the airspeed bled off, things got very interesting. Plenty of directional control in roll and yaw, but nothing for pitch. No matter what we did with the engine, we were just going down about 800FPM. I relieved the back pressure on the cyclic, and let off on the power, and it flew out of the 'stall'(yes I know it's not a stall) in full authority. Lesson learned - high power, back cyclic is a recipe for disaster, but,,,, that is precisely what's need for an in control landing profile. Back cyclic, and modest power. It is a devil to land cleanly unless one wants to spot land into a good breeze with zero airspeed. Even then, judging lift reserve, and descent profile takes finesse.
Moving forward, we went on our ways, and a few months later, he contacted me through the banner tow and asked if I would be interested in buying the thing. Or buying half of the thing, or buying shares in the thing. I had some cash in my pocket so we chatted for a while, and agreed to meet again for another flight. I wanted to see if it would go somewhere, and then come back. I suggested we fly up to Fallbrook which is about 40 miles N of Gillespie, and then come back. He agreed, but said he would do the landings. I said fine, makes sense to me. I took off, we turned right and found Fallbrook fine, but it was a 10kts crosswind. At first, he just wanted to circle and go back to Gillespie, but I wanted to land and do another take off. So he set up his approach, and lined up well, and here we came in for a landing with an 8-10kt crosswind. We shouldn't have, and gone back to Gillespie.
We touched down on the right main tire, and he throttled and yawed it some, to get the left tire down. It came down, but we were wiggling, and finally he just firewalled the engine, and we went around. Next pass was similar, with both wheels down, and speed coming down, and the right wheel lifts and we skid a bit sideways, and again he firewalls the engine and we lift off. I advised him to make an angled approach from the lee side of the runway, slightly crosswise to the center-line and see if that would help. He agreed, and we came in and touched down right at the edge of the runway pavement, headed about 30deg off-set to the runway heading into the wind. This time, he closed the throttle with both main wheels down, and more or less stabbed the brakes, thus pitching the nose gear on pretty firm. We were down, and I understood why he wanted to sell any or all of the plane.
Our trip back was uneventful, except we landed with about 4 gallons in the tank, maybe 30 min of reserve, but unlikely. I advised him I wasn't interested in the ship, paid for half the fuel we burned and wished him good luck. I only saw the plane once more a few months later, and have no idea what happened to it, but more than likely it was rolled somewhere and was not repaired. Thus endeth my time in the J2, about 1.3 hours total. I went on to fly both the B-8M, and more recently a GyroBee which used a 2 stroke 2 cyl engine and was the most twitchy thing I'd ever been in. But still not a handful like the J2.