Linear prediction (LP) is an ubiquitous analysis method in speech processing. Various studies have focused on sparse LP algorithms by introducing sparsity constraints into the LP framework. Sparse LP has been shown to be effective in several issues related to speech modeling and coding. However, all existing approaches assume the speech signal to be minimum-phase. Because speech is known to be mixed-phase, the resulting residual signal contains a persistent maximum-phase component. The aim of this paper is to propose a novel technique which incorporates a modeling of the maximum-phase contribution of speech, and can be applied to any filter representation. The proposed method is shown to significantly increase the sparsity of the LP residual signal and to be effective in two illustrative applications: speech polarity detection and excitation modeling.