There are a total of $ 2^6=64 $ equally likely sequences of heads and tails coin flips. Each toss corresponds to a movement clockwise or counterclockwise, so each sequence of coin tosses corresponds to a sequence of six movements, $ L $ or $ R $. If the man gets six consecutive heads or tails, corresponding to $ RRRRRR $ or $ LLLLLL $, then he will return to the starting point. But, the man could also flip three heads and three tails in some order, corresponding to a sequence like $ RRLRLL $. There are a total of $ \binom{6}{3}=20 $ sequences of moves that include three moves counterclockwise, and three clockwise. The probability that the man ends up where he started is: $$\frac{20+1+1}{64}=\boxed{\frac{11}{32}}$$