We present a visual computing framework for analyzing moral rhetoric on social media around controversial topics. Using Moral Foundation Theory, we propose a methodology for deconstructing and visualizing the \textit{when}, \textit{where}, and \textit{who} behind each of these moral dimensions as expressed in microblog data. We characterize the design of this framework, developed in collaboration with experts from language processing, communications, and causal inference. Our approach integrates microblog data with multiple sources of geospatial and temporal data, and leverages unsupervised machine learning (generalized additive models) to support collaborative hypothesis discovery and testing. We implement this approach in a system named MOTIV. We illustrate this approach on two problems, one related to Stay-at-home policies during the COVID-19 pandemic, and the other related to the Black Lives Matter movement. Through detailed case studies and discussions with collaborators, we identify several insights discovered regarding the different drivers of moral sentiment in social media. Our results indicate that this visual approach supports rapid, collaborative hypothesis testing, and can help give insights into the underlying moral values behind controversial political issues. Supplemental Material: https://osf.io/ygkzn/?view_only=6310c0886938415391d977b8aae8b749