So, if one is asked if a given matrix $A$ is symmetric, one could compute $A^T$ and check if $A^T=A$, however you can also simply check the symmetric entries accross the diagonal and see if they are equal (i.e whether $A_{ij}=A_{ji}$).
If one wishes to check if a square matrix is orthogonal, one could compute all the dot products of it's rows and columns, which, assuming an $n\times n$ matrix, requires $2\times\binom{n}{2}$ dot products (and this is without taking into account that as $n$ increases, so does the number of arguments in the sum of each dot product).
So, is there a shortcut that one may use to tell that a given matrix is orthogonal without having to compute all the dot products?
Computing the dot product is the same as computing the matrix product $A^TA$ and checking it is equa to the identity matrix $I$, that, with the naive algorithm, costs $O(n^3)$ FLOPS.
With fast matrix multiplications, though, you can get down to $O(n^{2.3728639})$ FLOPS, so you can gain something substantial in time.