The origin of and interactions between key thermodynamic anomalies are derived and analyzed, as are the interactions with the stability (or cavitation) limits. The conditions for interaction are derived from the underlying thermodynamic relations rather than using the more-commonly applied Taylor expansion method. As a result, we derive a general set of equations that govern the interactions between different lines of thermodynamic anomalies using standard manipulation of thermodynamic equations. The validity of the derivations is investigated by comparing them to numerical simulation data and previous Taylor expansion-based results. Simulations are performed using a modified Stillinger-Weber potential in which the balance of the two- and three-body interactions is varied and which serves to highlight the relationships between the various anomalies. The deeply supercooled regime is explored by employing replica exchange methods. The behavior of the anomalies is considered in terms of previously constructed thermodynamic "scenarios." Based on the newly uncovered interaction schemes, we propose a classification strategy for the thermodynamic anomalies (as first- or second-order) which could be extended to additional related anomalies.