The agent after 'by' becomes the subject, and the old subject becomes the object.

The reverse is the same three moves in the opposite direction, and one extra step: 'be' is deleted and the participle is put back into a finite tense.

The order was signed by the registrar. → The registrar signed the order.

the agent leads, 'by' disappears

The order signed the registrar.

the two nouns exchanged without changing the verb

Where it bites

The tense is carried by 'be', not by the participle. Read 'was' or 'has been' to find the tense before you touch the verb.