With both hooked up, you could turn on a switch, turn on with the remote, then turn off at switch or remote, but I don't think there is a way to turn it back on from the remote at that point, you would have to do a switch again in that sequence before the remote would work again. I guess if the remote triggered a 24 volt relay that is switched by 110, if such a thing exists, you could wire that in as one of the switches, again if they are 2 pole low voltage switches, and it might work. Lots of ifs there. I think though to use both, your switches would have to be a master on/off and the remote the cyclone on/off, which would be the safest way to wire a remote in my view.
The first picture is 220 huh? Did you have something 110 running off of it as well? The two black output wires connected to the one black input wire is confusing, as are the two ground wire inputs. Then the two ground wires go to a white wire. White is usually a neutral. Black is hot. Bare is ground. Red is hot in 220 wiring. If you had something 110 tagged on to the 220 circuit, it wasn't done right. There should have been a white input wire with the others to be done right. That's why there is 10-2 with ground (220 wiring) and 10-3 with ground (220 wiring with a neutral to be able to run 110 off the same circuit, say for a dryer or oven that uses 110 to run the timer) Yes, ground and neutral are bonded at the box, but are to be kept separate in the building's wiring. So it looks like you can use this as a 20 amp 220 tool outlet, but not as a 110 outlet.
Jim.