What is the name of the regime's secret police?

The regime's secret police are called the Eyes. Their fuller title is the Eyes of God, but “the Eyes” is the standard name used in the story and is the canonical answer to the question.

The Eyes operate as Gilead's intelligence and internal-security service. They monitor citizens, investigate suspected disobedience, make arrests, and create a climate in which almost anyone might be reporting to the authorities. Their black vans and sudden disappearances become symbols of the regime's hidden power.

The organization is deliberately difficult to understand from Offred's perspective. She cannot always tell whether a person is a loyal Gilead agent, a resistance member, or someone pretending to be one. That uncertainty makes surveillance an especially effective form of control.

The Eyes should not be confused with the Guardians, who are lower-ranking armed security personnel, or the Angels, who are soldiers. Nor are they the Salvagers, associated with public executions. In the television adaptation, Nick Blaine is revealed to have ties to the Eyes, adding another layer of suspicion to his relationship with June.