We clarify the theoretical foundations of partisan fairness standards for district-based democratic electoral systems, including essential assumptions and definitions that have not been formalized or in some cases even discussed. We pare assumptions down to their minimal essential components and add extensive empirical evidence for those with observable implications. Throughout, we follow a fundamental principle of statistics too often ignored -- defining the quantity of interest separately so its measures are vulnerable to being proven wrong, evaluated, and improved. This enables us to prove which approaches -- claimed in the literature to be estimators of partisan symmetry, the most widely accepted standard -- are statistically appropriate and which are biased, limited, or not measures of symmetry at all. Because real world redistricting involves complicated politics with numerous participants and conflicting goals, measures biased for partisan fairness sometimes still provide useful descriptions of other aspects of electoral systems.