Residual connections and normalization layers have become standard design choices for graph neural networks (GNNs), and were proposed as solutions to the mitigate the oversmoothing problem in GNNs. However, how exactly these methods help alleviate the oversmoothing problem from a theoretical perspective is not well understood. In this work, we provide a formal and precise characterization of (linearized) GNNs with residual connections and normalization layers. We establish that (a) for residual connections, the incorporation of the initial features at each layer can prevent the signal from becoming too smooth, and determines the subspace of possible node representations; (b) batch normalization prevents a complete collapse of the output embedding space to a one-dimensional subspace through the individual rescaling of each column of the feature matrix. This results in the convergence of node representations to the top-k eigenspace of the message-passing operator; (c) moreover, we show that the centering step of a normalization layer — which can be understood as a projection — alters the graph signal in message-passing in such a way that relevant information can become harder to extract. Building on the last theoretical insight, we introduce GraphNormv2, a novel and principled normalization layer. GraphNormv2 features a learnable centering step designed to preserve the integrity of the original graph signal. Experimental results corroborate the effectiveness of our method, demonstrating improved performance across various GNN architectures and tasks.