If both parties are going to give the tribunal a skeleton argument or written submissions, the usual thing is to exchange them simultaneously. The idea is that neither side gets to write their document having the unfair advantage of a preview of their opponent’s. This is broadly sensible and fair, but not something to get desperately worked up about: after all, you’re probably going to get the chance to say your piece orally anyway, and if not, you can always send in a supplementary document if you think there’s material in the other side’s that you haven’t dealt with and you must answer.
One exception to the usual rule is if there is an appeal to the EAT and a cross-appeal (see glossary), and a preliminary hearing at which the EAT proposes to decide whether either the appeal or the cross-appeal should have a full hearing. In that situation, strictly speaking you’re only there (whether you’re the original appellant, or the respondent to the appeal who is also cross-appealing) to argue that your own appeal merits a full hearing. So don’t be astonished if your opponent refuses to exchange skeletons with you: they may say you’re not entitled to it, because you’re not there to argue about their appeal at all, only your own.
This is probably technically right, if a bit arsey, so there’s not much point jumping up and down about it if is the line your opponent insists on taking. But the more sensible and co-operative course is to exchange skeletons so that if the judge wants to ask you about your opponent’s appeal (or vice versa), you have some idea what you’re talking about.