If the distribution is symmetric, then we have
p(x<32nd percentile) = a
p(x>68th percentile) = a
Notice the change of direction. The first is the left tail and the second is the right tail.
If it is true that that p(x < 32nd percentile) = p(x < 68th percentile), then it must also be the case that p(32nd percentile < x < 68th percentile) = 0. You've a very large hole in the middle of your distribution.
If there were some specifics, we wouldn't have to guess.