Vision and language models (VL) are known to exploit unrobust indicators in individual modalities (e.g., introduced by distributional biases) instead of focusing on relevant information in each modality. That a unimodal model achieves similar accuracy on a VL task to a multimodal one, indicates that so-called unimodal collapse occurred. However, accuracy-based tests fail to detect e.g., when the model prediction is wrong, while the model used relevant information from a modality. Instead, we propose MM-SHAP, a performance-agnostic multimodality score based on Shapley values that reliably quantifies in which proportions a multimodal model uses individual modalities. We apply MM-SHAP in two ways: (1) to compare models for their average degree of multimodality, and (2) to measure for individual models the contribution of individual modalities for different tasks and datasets. Experiments with six VL models -- LXMERT, CLIP and four ALBEF variants -- on four VL tasks highlight that unimodal collapse can occur to different degrees and in different directions, contradicting the wide-spread assumption that unimodal collapse is one-sided. Based on our results, we recommend MM-SHAP for analysing multimodal tasks, to diagnose and guide progress towards multimodal integration. Code available at \url{https://github.com/Heidelberg-NLP/MM-SHAP}.