Q:

Are space and time hierarchies even comparable?

I am wondering if there are any results to what extent the space and time hierarchies "disagree" on which problem is harder. For example, is it known whether there are languages $L_1$ and $L_2$ such that $L_1 \in \DeclareMathOperator{TIME}{TIME} \TIME(f(n))\setminus SPACE(g(n)),L_2\in \DeclareMathOperator{SPACE}{SPACE} \SPACE(g(n)) \setminus \TIME(f(n))$? How often does this occur?
P.S.- The question Function with space-depending computation time seems to ask something similar but was worded confusingly and none of the answers seem to be what I'm looking for.

A:

You can get the situation you describe by choosing weird functions $f(n)$ and $g(n)$. 
For example, let $g(n) = n^3$ and $$f(n) = \begin{cases}
 n & \text{if $n$ is odd},
\\\
 2^{n^5} & \text{if $n$ is even}.
\end{cases}
$$
Then choose $L_1$ and $L_2$ as follows:
$L_1$ is a language containing only strings of even length which can be decided in time $O(2^{n^5})$ but not in time $O(2^{n^4})$. The existence of such a language is pretty easy to prove from the time hierarchy theorem.
$L_2$ is a language containing only strings of odd length which can be decided in space $O(n^3)$ but not in space $O(n^2)$. The existence of such a language is pretty easy to prove from the space hierarchy theorem.
Then we have the following facts:
$L_1 \in TIME(f(n))$:
To decide whether a string is in $L_1$, simply check whether the length $n$ is even. If it is, then continue to use the $O(2^{n^5})$ time decider for $L_1$ whose existence is guaranteed by the definition of $L_1$. If $n$ is odd, immediately reject since $L_1$ does not include any odd length strings anyway. This procedure decides $L_1$, runs in time $O(n)$ when $n$ is odd, and runs in time $O(2^{n^5})$ when $n$ is even. In other words, this procedure decides $L_1$ in time $O(f(n))$. As desired, $L_1 \in TIME(f(n))$.
$L_2 \in SPACE(g(n))$:
By the definition of $L_2$, $L_2$ can be decided in space $O(n^3)$. Thus, $L_2 \in SPACE(n^3) = SPACE(g(n))$, as desired.
$L_1 \not\in SPACE(g(n))$:
Suppose for the sake of contradiction that $L_1 \in SPACE(g(n)) = SPACE(n^3)$. We know that $SPACE(n^3) \subseteq TIME(2^{O(n^3)}) \subsetneq TIME(2^{n^4})$. Thus, there exists a decider for $L_1$ which runs in time $O(2^{n^4})$. This directly contradicts the definition of $L_1$. Then by contradiction, we see that $L_1 \not\in SPACE(g(n))$.
$L_2 \not\in TIME(f(n))$:
Suppose for the sake of contradiction that $L_2 \in TIME(f(n))$. This means that there exists a constant $c$ and an algorithm $A$ deciding $L_2$ such that on any input of size $n$, algorithm $A$ terminates in time $c\times f(n)$. 
We construct a new algorithm $A'$ as follows: given some input, walk through the entire input, keeping track of whether the input length is even or odd; if at the end of the input the length is determined to be odd, return to the start of the input and run $A$; otherwise, reject. For any input of odd length, $A'$ returns the same answer as $A$. For any input of even length, $A'$ rejects, which matches the expected behavior since $L_2$ contains no even length strings. Thus, $A'$ also decides $L_2$. On even length inputs, $A'$ runs for exactly $n$ steps. On odd length inputs, $A'$ runs for exactly $2n$ steps more than $A$ requires. But $A$ requires at most $c\times f(n)$ steps, which for odd $n$ is $cn$. Thus, in all cases, $A'$ runs in at most $(c+2)n$ steps. In other words, algorithm $A'$ decides $L_2$ in time $O(n)$. 
But since $TIME(n) \subseteq SPACE(n)$, we can conclude that $L_2 \in SPACE(n) \subsetneq SPACE(n^2)$. This contradicts the definition of $L_2$. Thus, by contradiction we see that $L_2 \not\in TIME(f(n))$.

