Multimodal representation learning techniques typically require paired samples to learn shared representations, but collecting paired samples can be challenging in fields like biology, where measurement devices often destroy the samples. This paper presents an approach to address the challenge of aligning unpaired samples across disparate modalities in multimodal representation learning. We draw an analogy between potential outcomes in causal inference and potential views in multimodal observations, allowing us to leverage Rubin's framework to estimate a common space for matching samples. Our approach assumes experimentally perturbed samples by treatments, and uses this to estimate a propensity score from each modality. We show that the propensity score encapsulates all shared information between a latent state and treatment, and can be used to define a distance between samples. We experiment with two alignment techniques that leverage this distance---shared nearest neighbours (SNN) and optimal transport (OT) matching---and find that OT matching results in significant improvements over state-of-the-art alignment approaches in on synthetic multi-modal tasks, in real-world data from NeurIPS Multimodal Single-Cell Integration Challenge, and on a single cell microscopy to expression prediction task.