Q:

Are reflections in higher dimensions commutative?

If $~r_1, r_2 ,\cdots, r_t~$ are a sequence of reflections about $~x_1=\frac{1}{2},\cdots,x_n=\frac{1}{2}~$ respectively in n dimensions does the sequence of reflections matter?

A:

Absolutely! The order matters.
In $\Bbb{R}^n$ the orthogonal reflection $s_{ij}$ with respect to the hyperplane $x_i=x_j$ interchanges the two involved coordinates: 
$$s_{ij}:(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_n)\mapsto
(x_1,\ldots,x_{i-1},x_j,x_{i+1},\ldots,x_{j-1},x_i,x_{j+1},\ldots,x_n).
$$
In other words, it permutes the coordinates according to the 2-cycle $(ij)$.
But from an encounter with permutation groups you probably remember that 2-cycles don't always commute. For they generate the entire symmetric group $S_n$, which is non-commutative when $n\ge3$.

As Lee Mosher explained, an even simpler case of non-commuting reflections is with respect to two parallel hyperplanes. In that case the commutator of the two reflections is a translation.
But, if two hyperplanes, $H_1$ and $H_2$, in $\Bbb{R}^n$ are orthogonal to each other (= if their respective normal vectors are orthogonal), then they do commute.

