Short algebraic verification.

Write out the triple product in components:

$$\vec a\cdot(\vec b\times\vec c) = a_1(b_2c_3-b_3c_2) + a_2(b_3c_1-b_1c_3) + a_3(b_1c_2-b_2c_1).$$

If you cyclically permute $(a,b,c)\mapsto(b,c,a)$ you obtain

$$\vec b\cdot(\vec c\times\vec a) = b_1(c_2a_3-c_3a_2) + b_2(c_3a_1-c_1a_3) + b_3(c_1a_2-c_2a_1).$$

By explicit rearrangement (or using multilinearity and antisymmetry of the crosproduct) these two expressions are identical. Thus the scalar triple product is invariant under cyclic permutations.