Video solution:

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Written solution:

A permutation is valid exactly when the four comparison signs between consecutive terms alternate. Thus the signs must be either up-down-up-down or down-up-down-up.

For the up-down-up-down pattern, a direct count by the middle value, or equivalently the standard alternating-permutation count for five distinct numbers, gives $16$ permutations. Replacing every entry $x$ by $6-x$ gives a bijection from these to the down-up-down-up permutations, so there are another $16$.

The total number of valid rearrangements is $16+16=32$.

Thus, D is the correct answer.