Here we are in the superior (supreme) courts, i.e. appealing against a decision handed down by an appellate court in this case, so the terminology changes. In an appeal, there is the Appellant ( i.e. the party lodging the appeal againt the finding of a lower court) and the Respondent ( i.e. the party defending the decision). This proceeding is is where the respondent is appealing against a decision handed down by an appeal court and so becomes the cross petitioner or cross appellant. This is probably an administrative process (public contract tenders, public competitions for teaching positions, etc.).
The idea is this. While the cross petitioner is seeking to reverse a judgment based on a series of main claims or on the facts of the case or in law, the nominal petitioner is seeking to reverse the judgment based on new elements to the case that are relevant to the main claims but which are specifically to his/her advantage and not necessarily to that of the cross petitioner.
Should the court reject the main claims of the cross petitioner's appeal then the nominal parties claims will also be rejected as a result. In practice, the nominal petitioner's claims are examined before the cross petitioner's main arguments.
Alison