If you want to describe inversion symmetric systems the Hamiltonian matrix is a symmetric matrix and otherwise it is a Hermitian matrix. This is an important property to obtain real-valued energy eigenvalues.
Depending on the approach to density functional theory this property of the Hamiltonian matrix is not mathematically guaranteed in the first place but has to be ensured by postprocessing measures like the one you describe in the question.
Let us assume that you are using a (linearized) augmented-plane-wave basis set as it is used in software packages like Wien2k, Fleur, Elk, Exciting and others. With such a representation the unit cell is separated into non-overlapping but nearly touching so-called muffin-tin (MT) spheres around each atom $\alpha$ and an interstitial region (IR) in between. In the IR such a basis function is a plane wave with wave vector $\mathbf{k} + \mathbf{G}$, where $\mathbf{k}$ is a <PERSON> vector and $\mathbf{G}$ a reciprocal lattice vector. In the MT spheres each basis function consists of radial functions times spherical harmonics up to a certain angular momentum cutoff parameter $l_\text{max}$. 976
The radial functions are scaled to ensure continuity in value (and slope) at the MT spheres boundaries. But this matching of course only happens for $l \le l_\text{max}$. For higher $l$ the basis functions feature discontinuities at the MT sphere boundaries.
Let us test whether we obtain a Hermitian matrix with such a basis. Hermiticity means
\begin{equation} H_{\mathbf{G}\mathbf{G}'}^{\mathbf{k}} - \left(H_{\mathbf{G}'\mathbf{G}}^{\mathbf{k}}\right)^\ast = 0. \end{equation}
With the sketched basis consisting of IR representations $\phi_{\mathbf{k}\mathbf{G}}^\text{IR}$ and MT representations $\phi_{\mathbf{k}\mathbf{G}}^\alpha$ this expression becomes \begin{align} H_{\mathbf{G}\mathbf{G}'}^{\mathbf{k}} - \left(H_{\mathbf{G}'\mathbf{G}}^{\mathbf{k}}\right)^\ast & = \left\langle \phi_{\mathbf{k}\mathbf{G}}^\text{IR} \middle\vert \hat{H} \middle\vert \phi_{\mathbf{k}\mathbf{G}'}^\text{IR} \right\rangle_\text{IR} - \left(\left\langle \phi_{\mathbf{k}\mathbf{G}'}^\text{IR} \middle\vert \hat{H} \middle\vert \phi_{\mathbf{k}\mathbf{G}}^\text{IR} \right\rangle_\text{IR} \right)^\ast \nonumber \ & \phantom{=} + \sum\limits_\alpha \left\langle \phi_{\mathbf{k}\mathbf{G}}^\alpha \middle\vert \hat{H} \middle\vert \phi_{\mathbf{k}\mathbf{G}'}^\alpha \right\rangle_{\text{MT}^\alpha} - \left(\left\langle \phi_{\mathbf{k}\mathbf{G}'}^\alpha \middle\vert \hat{H} \middle\vert \phi_{\mathbf{k}\mathbf{G}}^\alpha \right\rangle_{\text{MT}^\alpha} \right)^\ast \nonumber \ & = -\frac{1}{2} \left\lbrace\int\limits_\text{IR} \left(\phi_{\mathbf{k}\mathbf{G}}^\text{IR}(\mathbf{r})\right)^\ast \nabla^2 \phi_{\mathbf{k}\mathbf{G}'}^\text{IR}(\mathbf{r}) - \phi_{\mathbf{k}\mathbf{G}'}^\text{IR}(\mathbf{r}) \nabla^2 \left(\phi_{\mathbf{k}\mathbf{G}}^\text{IR}(\mathbf{r})\right)^\ast d^3r \right. 677