Electronman, I think I misunderstood your previous schematic. I thought your control signals were square waves, with the low level being 0 and the high levels being +5 for one of the controls and -5 for the other. i now understand that they are sine waves, 10V p-p, and 180 degrees out of phase. My bad.
However, they still need to be square waves that go from -6V to +6V, or at least -5V to +5V.
The comment about the switch being half off (or on) was due to my misunderstanding about your control signals, which I thought were at zero volts for half of each cycle.