NPSP reciprocal relationship returns incorrect inverse value for custom relationship types

In NPSP, reciprocal relationship types are managed through Custom Settings > Relationship Lookup Settings. When a relationship is created between two Contacts, NPSP automatically creates an inverse relationship using the configured reciprocal value.
In some orgs, specific reciprocal relationship types — particularly those involving gendered or extended family terms (e.g., Parent-in-Law/Child-in-Law, Step-parent/Step-child) — may fail to apply the correct inverse value. Instead of creating the expected paired relationship, both sides of the relationship may display the same label (e.g., both sides showing "Parent-in-Law" instead of one showing "Child-in-Law").
This issue has been reported when:

• Custom Relationship Lookup Settings have been in place for several years without modification
• New relationships of specific types are added after an NPSP package upgrade
• Certain relationship type pairs that rely on inverted string matching behave differently than other pairs (e.g., parent/child works correctly but parent-in-law/child-in-law does not)