This paper introduces a novel Laplacian matrix aiming to enable the construction of spectral convolutional networks and to extend the signal processing applications for directed graphs. Our proposal is inspired by a Haar-like transformation and produces a Hermitian matrix which is not only in one-to-one relation with the adjacency matrix, preserving both direction and weight information, but also enjoys desirable additional properties like scaling robustness, sensitivity, continuity, and directionality. We take a theoretical standpoint and support the conformity of our approach with the spectral graph theory. Then, we address two use-cases: graph learning (by introducing HaarNet, a spectral graph convolutional network built with our Haar-Laplacian) and graph signal processing. We show that our approach gives better results in applications like weight prediction and denoising on directed graphs.