Just as a reference, let's cite the M-W's definition of the words:
- "used to introduce statements that describe two different or opposite ideas, people, etc."
For some reason, I feel like it is also correct and idiomatic to use the phrases when the two parts are just two different aspects of something, but not necessarily opposing:
@StoneyB in his great contribution to ELL showed that present aspect is a very interesting yet involved area of English language[Ref]. On the other hand, we know that there are many posts regarding the delicacies of conditional sentences. Thus, the combination of perfect aspect with conditional sentences would an interesting area to explore!
I mean, I think we can also use "on the other hand" to introduce another related thing which is not necessarily in contrast to the stated or implied "on the one hand"!