In court, a legal representative may be conducting the case and calling witnesses and cross examining the other side's witnesses.
Are these representatives "sworn in" under oath?
As they may also deliver submissions of legal arguments, if not, perhaps, statements of fact, can they also be cross-examined by the counterparty?
In the united-states, attorneys are almost never placed under oath. Their statements -- whether they are assertions of fact or legal arguments -- are not evidence, and they are not subject to cross examination.
There are limited circumstances in which an attorney would be competent to present actual evidence. In such cases, the attorney would be sworn in as a witness and subject to cross examination, but the attorney would likely also be disqualified from acting as an attorney in such a situation under Model Rule 3.7.
Only witnesses are cross-examined. See various rules of procedure: e.g. Ontario, British Columbia.
The procedure for responding to counsel submissions of their legal arguments is to make submissions in reply. These would be done at opening and closing.
As such, they are not subject to the rules of evidence including cross-examination.
When they argue to law they are making submissions, not giving evidence. In general, a wise judge asks them to make their submissions to each other before submitting them - most sensible attorneys actually agree on 99% of what the law is if they listen to each others points.
