How would I go about showing that L $\leq_p$ L' does not necessarily imply L' $\leq_p$ L? I was thinking I should show an example of two problems, where one can reduce to the other but not the other way round, but am not sure what such problems could be.
You have a good plan of approach. Now you need to spend some more time pursuing this. Try to come up with a list of languages $L$ you've seen before, and try different pairs to see if they satisfy the condition. After some trial and error you should be able to solve your problem.
Maybe this insight will be helpful (along with your own thoughts and D.W.'s answer):
Quite intuitively, the relation $A \le_p B$ indicates that $B$ is more "difficult" than $A$. Indeed, for an NP-hard problem, like SAT, we know that all other NP languages $A$ can be reduced to it, which makes it "the hardest".. But not all languages are NP-hard, some languages are "easier" than others..