What is the name for a process that can be reversed without increasing total entropy?

A process that can be reversed without increasing total entropy is called a reversible process.

In thermodynamics, reversibility means that both a system and its surroundings can be returned to their original states by an infinitesimal change in conditions. The total entropy change of the system plus surroundings is zero. Such a process must proceed quasistatically, keeping the system very close to equilibrium, and must avoid friction, turbulence, and other dissipative effects.

Reversible processes are idealizations rather than ordinary real-world events. They provide a benchmark for maximum efficiency: the Carnot cycle, for example, represents the highest possible efficiency for a heat engine operating between two temperatures. Real engines always produce some entropy because heat transfer occurs across finite temperature differences and mechanical parts experience losses.

A common mix-up is confusing “reversible” with “cyclic.” A cyclic process merely returns the system to its starting state; it may be reversible or irreversible. Likewise, a quasistatic process is not automatically reversible if friction or another source of dissipation is present.