No permutation is both odd and even. $(123)$ is an even permutation. It is the cycle that sends $1\mapsto 2\mapsto 3\mapsto 1$. It is not the identity permutation. This cycle notation may be a bit confusing in this way if we also use two line notation, in that we also write the two line notation with parentheses and it means something completely different. I usually see one line notation without parentheses, so $123$ is the identity permutation, but $(123)$ is a cycle with an even number of inversions.
Note that you can write $(123) = (312) = (231)$. So your method of detecting inversions is not correct. An inversion in the cycle does not correspond to an inversion in the permutation.
In general a cycle of length $2k$ is an odd permutation, and a cycle of length $2k+1$ is even. This is a pretty simple rule. If you write a permutation as a product of disjoint cycles, the parity is additive as one would expect, as is true for any product of permutations. An easy way to remember this is as follows:
$$(123) = (12)(23)$$
$$(2341) = (23)(34)(41) = (23)(34)(14)$$
You can in general split a cycle into a product of transpositions this way, and the number of transpositions, while not the number of inversions, has the same parity as such.
By the definition of a cycle, it is not terribly difficult to prove this multiplication rule. You should give it a shot. Even more fun, we have
$$(125347) = (125)(5347) = (125)(534)(47)$$
etc. This splitting rule is a rule I find very useful.
As a warning, if you multiply permutations in the opposite order, as in not according to function composition, the pretty splitting rule disappears. Then you'd have
$$(123) = (23)(12)$$
This to me is evidence that multiplication in that order is unnatural, but it may have some advantages that I'm not aware of. I find this is sufficient reason not to use it for my purposes, but of course if it is what you use in your course or textbook, it is what it is. You can still use my splitting rule, but you have to reverse the order. You can recover the naturality of the splitting rule by interpreting cycles in the opposite order, but as far as I know this is not done.