Can a witness examined by one party be cross-examined by the other?

This does not demonstrate bias by the court. "A witness examined by one party may be cross-examined upon any matter within the scope of the direct examination by each other party to the action in such order as the court directs." (Evid. Code, 773, subd. (a).) "'Cross-examination is the principal means by which the believability of a witness and the truth of his testimony are tested. . . . [T]he cross-examiner is not only permitted to delve into the witness' story to test the witness' perceptions and memory, but the cross-examiner has traditionally been allowed to impeach, i.e., discredit, the witness.' [Citation.]" (City of Los Angeles v. Superior Court (2002) 29 Cal.4th 1, 23.)