More generally, if $1\le k\le N-1$ is an integer, where $N$ is a positive interger, $$S_{N,k} := \sum_{n=0}^\infty\biggl( \frac{1}{(N n-k)^2} + \frac{1}{(N n+k)^2} \biggr) = \frac{\pi^2}{N^2\sin^2(\pi k/N)}.$$ Your sum is $S_{10,1}-S_{10,3}$.

More generally, if $1\le k\le N-1$ is an integer, where $N$ is a positive interger, $$S_{N,k} := \sum_{n=0}^\infty\biggl( \frac{1}{(N n+N-k)^2} + \frac{1}{(N n+k)^2} \biggr) = \frac{\pi^2}{N^2\sin^2(\pi k/N)}.$$ Your sum is $S_{10,1}-S_{10,3}$.

More generally, if $1\le k\le 9$ is an integer, $$S_k := \sum_{n=0}^\infty\biggl( \frac{1}{(10n-k)^2} + \frac{1}{(10n+k)^2} \biggr) = \frac{\pi^2}{100\sin^2(\pi k/10)}.$$ Your sum is $S_1-S_3$.

More generally, if $1\le k\le N-1$ is an integer, where $N$ is a positive interger, $$S_{N,k} := \sum_{n=0}^\infty\biggl( \frac{1}{(N n-k)^2} + \frac{1}{(N n+k)^2} \biggr) = \frac{\pi^2}{N^2\sin^2(\pi k/N)}.$$ Your sum is $S_{10,1}-S_{10,3}$.