Yes, the problem is NP-complete. Here is a reduction of 1-in-3 SAT to your problem.
Let $\{C_i:i< n\}$ be the given collection of 3-clauses in variables $\{x_j:j< m\}$. All tiles and the region will have length $n+m+2$, so no shift is possible. For each of the $2m$ literals $a$ (which is $x_j$ or $\neg x_j$), we consider a tile of the form
$$1,\underbrace{0,\dots,0,1,0,\dots,0}_m,u_{a,0},\dots,u_{a,n-1},0,$$
where the $1$ in the left part appears at position $j+1$, and $u_{a,i}$ is $1$ if $a$ occurs in $C_i$, and $0$ otherwise. The region is
$$m,\underbrace{1,1,\dots,1}_{n+m},0.$$
Note that the first and last number prevent reversal from being used. Then the left part forces any possible sum to use exactly one of the two literals $x_j$, $\neg x_j$ for any $j$, and the right part ensures that each clause is satisfied by exactly one of the selected literals.
Another variant of the reduction is that we consider tiles and the region of length $2n$; for each $j< m$, we take the tile
$$u_{x_j,0},u_{x_j,1},\dots,u_{x_j,n-1},u_{\neg x_j,n-1},u_{\neg x_j,n-2},\dots,u_{\neg x_j,0},$$
and the region is
$$\underbrace{1,1,\dots,1}_{n},\underbrace{2,2,\dots,2}_{n}.$$
Again, no shift is possible. Moreover, any solution must use all of the tiles, so the only choice is whether we reverse a given tile or not, which selects the variable assignment.
