A "classic" way to do this is using the signature; another way is computing Jones polynomial; a third way is using Heegaard Floer homology, and the $\tau$-invariant of Ozsváth and Szabó: they are all sensitive to orientation reversal in many cases.

For example, all three invariants tell apart the left-handed trefoil from the right-handed trefoil.

If you want to show that a knot is amphicheiral (i.e. isotopic to its mirror image), you have to fiddle around with diagrams and Reidemeister moves.

As far as I know, there is no general way of doing it, though.

There are several ways to try and tell apart a knot from its mirror

• Probably the most "classic" way to do this is using the signature;

• You can compute Jones polynomial and check that it's not symmetric;

• You can use Rasmussen's $s$-invariant coming from Khovanov homology;

• You can use Heegaard Floer homology, and the $\tau$-invariant of Ozsváth and Szabó.

Each of the invariants above is often sensitive to orientation reversal; for example, all four invariants tell apart the left-handed trefoil from the right-handed trefoil.

If you want to show that a knot is amphicheiral (i.e. isotopic to its mirror image), you have to fiddle around with diagrams and Reidemeister moves.

As far as I know, there is no general criterion to answer your question, though.