The in-in formalism is a systematic method for calculating primordial correlation functions in cosmology. However, the treatment of total time derivative interactions, often mistakenly considered as "boundary terms," has been a source of confusion and conceptual errors. This paper clarifies the role of these interactions and shows that they can lead to large, cancellable contributions in the in-in perturbation theory. The paper discusses the treatment of total time derivative interactions in the Lagrangian path integral formulation of the in-in perturbation theory and highlights the importance of terms proportional to the linear equations of motion. A new approach is proposed that avoids the use of integration by parts by using canonical transformations to simplify interactions in the full Hamiltonian. This method allows for the direct calculation of correlation functions in terms of phase-space variables, avoiding the complications of total time derivatives. The paper applies this method to single-field inflation with canonical kinetic terms, showing how it provides a clearer interpretation and computational simplification compared to the in-in method with total time derivatives. The results are important for performing complete calculations of exchange diagrams in inflation, such as the (scalar and tensor) exchange trispectrum and the one-loop power spectrum. The paper also discusses the importance of these results for understanding the non-linear dynamics of quantum states during inflation. The paper is structured into sections that cover the in-in formalism with total time derivatives, canonical transformations, single-field inflation, and conclusions. The paper includes appendices that provide additional material on canonical transformations, diagrammatic rules, and Lagrangian interactions.The in-in formalism is a systematic method for calculating primordial correlation functions in cosmology. However, the treatment of total time derivative interactions, often mistakenly considered as "boundary terms," has been a source of confusion and conceptual errors. This paper clarifies the role of these interactions and shows that they can lead to large, cancellable contributions in the in-in perturbation theory. The paper discusses the treatment of total time derivative interactions in the Lagrangian path integral formulation of the in-in perturbation theory and highlights the importance of terms proportional to the linear equations of motion. A new approach is proposed that avoids the use of integration by parts by using canonical transformations to simplify interactions in the full Hamiltonian. This method allows for the direct calculation of correlation functions in terms of phase-space variables, avoiding the complications of total time derivatives. The paper applies this method to single-field inflation with canonical kinetic terms, showing how it provides a clearer interpretation and computational simplification compared to the in-in method with total time derivatives. The results are important for performing complete calculations of exchange diagrams in inflation, such as the (scalar and tensor) exchange trispectrum and the one-loop power spectrum. The paper also discusses the importance of these results for understanding the non-linear dynamics of quantum states during inflation. The paper is structured into sections that cover the in-in formalism with total time derivatives, canonical transformations, single-field inflation, and conclusions. The paper includes appendices that provide additional material on canonical transformations, diagrammatic rules, and Lagrangian interactions.