As shown in this post:
We know that for any homotopy from the Identity to itself $F: X\times I \rightarrow X$
For any $x_0 \in X$,
the class of the loop $F(x_0, \cdot ):I\rightarrow X$ is in the center of the fundamental group $\pi_1(X, x_0)$
Is it possible to describe every element of the center in a similar way? (I'm guessinng not but can't come up with a counter-example)
If not, is there any known "nice" (and non trivial) property that caracterize those elements?