The computation of Moran’s index of spatial autocorrelation requires the definition of a spatial weighting matrix. The eigendecomposition of this doubly centered matrix (i.e., one that forces the sums of all rows and columns to equal zero) has interesting properties that have been exploited in various contexts: distribution properties of the Moran coefficient (MC), spatial filtering in linear models, generalized linear models, and multivariate analysis. In this article, this eigendecomposition is used to propose a new view of MC based on its interpretation in the simple context of linear regression. I use this interpretation to demonstrate the different properties of MC and also the inefficiency of this index in some situations involving simultaneous positive and negative spatial autocorrelation. I propose some new statistics and procedures for testing spatial autocorrelation, and conduct a simulation study to evaluate these new approaches.