The underlying working mechanism of Local Binary Pattern (LBP) techniques is still a topic to investigate. In this paper we explore symmetry & group theory for grouping functions that represent binary intensity changes obtained with the LBP technique. This additionally offers a strong mathematical foundation for the basis of the technique. We include complement and mirror invariants to the known LBP rotational invariant. We tested our algorithm using 13 textures from the Brodatz database. The statistical analysis shows that combining rotational, mirrored and complemented versions of local texture results in an improvement in the performance of the technique in terms of accuracy describing textures and discrimination distinguishing textures.