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Thus, a person called to testify before Congress can refuse if they believe the government could use the testimony against them in criminal proceedings. To render the protections against self-incrimination ineffective, Congress can provide immunity from criminal prosecution.
A defendant could conceivably be called to testify first, before the plaintiff testifies, in the plaintiff's case in chief. Sometimes, third party witnesses can be called out of order if their availability is limited. See California Evidence Code Section 776.