The method used in several papers by the present author to investigate the symmetry of distribution of various arithmetic functions \(f(n)\) about \(x\) in short intervals \([x- h,x+ h]\) is to study the “symmetry integral” \[ I_f(N, h):= \sum_{N< x\leq 2N}\,\Biggl| \sum_{|n-x|\leq h} \text{sgn}(n- x)f(n)\Biggr|^2,\tag{\(*\)} \] where \(h= o(N)\) as \(N\to\infty\) and \(\text{sgn}(t)={1\over|t|}\) if \(t\neq 0\). This approach has its origins in a paper [J. Math. Soc. Japan 45, No. 3, 447–458 (1993; Zbl 0787.11038)] by J. Kaczorowski and A. Perelli. The objective of the current paper is to study \((*)\) when \(f(n)\) is the divisor function \(\sigma_{-s}(n)= \sum_{d|n| d^{-s}}\), where \(s\in\mathbb{C}\), \(\text{Re}(s) =\sigma> 0\). Using an asymptotic version of the large sieve, the author establishes in Theorem 1 a formula for \(I_s(N,h)\) (given by \((*)\) with \(f= \sigma_{-s}\)) with a remainder term that is \(o(N)\) as \(N\to\infty\) under the assumption that \(h= N^\theta\) with \(0<\theta<\min({1\over 2},{\sigma\over 2+\sigma})\), \(\sigma> 0\). When also \(\theta< {1\over 2(2+\sigma)}\), it is shown in Corollary 1 that \[ I_s(N,h)= 2{|\zeta(1+ s)|^2\over \zeta(2+ 2\sigma)} N\eta^{(h)}(2\sigma)+ o(N), \] where \(n^{(h)}(2\sigma)\) is given by an infinite series depending on \(h\) that converges in \(\sigma> 0\).