Does it ever make sense to use the sum of the pre and post measure means? Say I quantitatively measure a specific trait of each of the students in a group so that each of the students receive a score. The average score of the group is $X_0$. After a treatment, I measure a gain, and the average score of the group post-treatment is $X_1$.
In most cases, what's interesting is the difference between $X_0$ and $X_1$ (analyzing the effectiveness of the treatment), but does it ever make sense to use the sum of $X_0$ and $X_1$ for any non-trivial purposes? If so, what is the meaning of $X_0 + X_1$?
In addition, any references concerning this topic would be greatly appreciated!!

Following a hint by @FrankHarrell, I look up recovery of inter-block information as one application of the sum of scores $X_0 + X_1$ as opposed to the difference of scores $X_1 - X_0$.
This is a classical concept in incomplete block design, proposed by Yates in [1]. For illustration I use an example by Portnoy in [2] where the "blocks" are students just as in the OP's example.

Source: (Portnoy, 1973)
This experiment compares the grading methods of four teachers / treatments {$T_1$, $T_2$, $T_3$, $T_4$}. Each teacher grades a composition by three students / blocks {$S_1$, ..., $S_{12}$} in two classes / replications (freshmen & senior). The design is incomplete (the teachers don't grade all the students) and balanced (each student is graded by two teachers and each teacher grades half of the freshmen and half of the seniors).
Since the design is incomplete, the comparisons of block totals contain information about the differences between treatments in addition to the direct comparisons between treatments within blocks. Since the design is balanced, it's possible but tedious to do the derivations by hand (see the references).
Let $X_{ij}$ denote the grade of student $S_i$ by teacher $T_j$. Each grade $X_{ij}$ is the sum of a fixed teacher effect $\beta_j$, a random student effect $u_i$ and an error term.
$$
\begin{aligned}
S_{1} &= X_{11} + X_{12} = \left(\beta_1 + u_1 + e_{11}\right) + \left(\beta_2 + u_1 + e_{12}\right) \\
S_{2} &= X_{22} + X_{23} = \left(\beta_2 + u_2 + e_{22}\right) + \left(\beta_3 + u_2 + e_{23}\right) \\
S_{1} - S_{2} &=
\left(\beta_1 + \beta_2\right) - \left(\beta_2 + \beta_3\right) + \left(2u_1 - 2u_2\right) + \left(e_{11} + e_{12} - e_{22} - e_{23}\right)
\end{aligned}
$$
The students / blocks are modeled as random effects (they are assumed to be drawn from a population of students) so the $u_i$s have mean 0. The errors have mean 0 as well. Therefore, comparisons of block totals contain information about treatment differences.
$$
\begin{aligned}
\operatorname{E}\left(S_{1} - S_{2}\right) = \beta_1 - \beta_3
\end{aligned}
$$
Note the block totals are not of interest themselves. Their purpose is to increase the precision (and thus increase the power) to estimate the treatment differences.
Otherwise the totals are not very interpretable unless there are no treatment effects: $E\left(S_i\right)$ is the aggregate grade for writing two compositions and being graded by two specific teachers. With nonzero teacher effects, a student will get different grades for the same compositions if they happen to be graded by other teachers.
[1] F. Yates. The recovery of inter-block information in variety trials arranged in three-dimensional lattices. Annals of Eugenics(?!), 9(2):136–156, 1939. 
[2] S. Portnoy. On recovery of intra-block information. Journal of the American Statistical Association, 68(342):384–391, 1973. 
