For simplicity, let's initially "fix" the rod (it can't be rotated 180 degrees) and label the bands a, b, c, d, e.
There are $2^5 = 32$ ways to color the five bands with two colors. Now we must count how many of them are "distinct".
Some of the colorings are symmetric - if you rotate the rod 180 degrees, the new coloring will coincide with the initial one. To completely determine a symmetric coloring, you need to know the coloring of a, b, c; so there are 8 symmetric colorings.
The remaining 24 colorings fall into pairs - if you "rotate the rod 180 degrees" starting with one of these colorings, you get a different coloring. So, out of these 24 colorings, only 12 are considered "distinct" under the definition of "distinct" as clarified in the OP's comment.
Adding back the symmetric colorings, there are 20 distinct colorings overall.