B cells surveil the body for foreign matter using their surface-expressed B cell antigen receptor (BCR), a tetrameric complex comprising a membrane-tethered antibody (mIg) that binds antigens and a signaling dimer (CD79AB) that conveys this interaction to the B cell. Recent cryogenic electron microscopy (cryo-EM) structures of IgM and IgG isotype BCRs provide the first complete views of their architecture, revealing that the largest interaction surfaces between the mIg and CD79AB are in their transmembrane domains (TMDs). These structures support decades of biochemical work interrogating the requirements for assembly of a functional BCR and provide the basis for explaining the effects of mutations. Here we report a focused saturating mutagenesis to comprehensively characterize the nature of the interactions in the mIg TMD that are required for BCR surface expression. We examined the effects of 600 single-amino-acid changes simultaneously in a pooled competition assay and quantified their effects by next-generation sequencing. Our deep mutational scanning results reflect a feature-rich TMD sequence, with some positions completely intolerant to mutation and others requiring specific biochemical properties such as charge, polarity or hydrophobicity, emphasizing the high value of saturating mutagenesis over, for example, alanine scanning. The data agree closely with published mutagenesis and the cryo-EM structures, while also highlighting several positions and surfaces that have not previously been characterized or have effects that are difficult to rationalize purely based on structure. This unbiased and complete mutagenesis dataset serves as a reference and framework for informed hypothesis testing, design of therapeutics to regulate BCR surface expression and to annotate patient mutations.