There are several different characterizations of permutations that are equivalent:
- Are composed of an even number of transpositions
- Are composed of cycles of odd number of elements
- Are member of the alternating group $A_n$
- Matrix representation has positive determinant
- Preserve orientation
If you're viewing the permutation group as acting geometrically, the last one is more salient than otherwise. If you look at where all the points are relative to each other, the identity and the two 120° rotations leave these relationships unchanged. Suppose you form a triangle on a clock face, with a red dot at 12, green at 4, and blue at 8. The three even permutations result in a configuration such that going around clockwise from red, you get green and then blue. The odd permutations result in configurations in which, starting at red and going clockwise, you have blue and then green.
Equivalently, the identity and the two 120° rotations can all be performed within the space that the triangle is embedded in; a triangle can be rotated while remaining in the plane. To physically reflect a triangle, however, requires rotation it through three dimensional space, rather than just the plane. In general, permutations of $n$ elements can be represented as actions on vertices of an $n-1$ regular simplex in $n-1$ space, where even permutations correspond to rigid transformations within that $n-1$ space, while odd permutations correspond to permutations that require $n$ dimensions to perform.