Turns in Hamilton cycles of rectangular grids

For a Hamilton cycle in a rectangular $m\times n$ grid, what is the greatest number of turns that can occur? We give the exact answer in several cases and an answer up to an additive error of $2$ in all other cases. In particular, we give a new proof of the result of Beluhov for the case of a square $n\times n$ grid. Our main method is a surprising link between the problem of ‘greatest number of turns’ and the problem of ‘least number of turns’.