The statute and rule are clear as donkey dung.

35 USC 134(a)
An applicant for a patent, any of whose claims has been twice rejected, may appeal from the decision of the primary examiner to the Board of Patent Appeals and Interferences, having once paid the fee for such appeal.

37 CFR 41.31
Who may appeal and how to file an appeal. (1) Every applicant, any of whose claims has been twice rejected, may appeal from the decision of the examiner to the Board by filing a notice of appeal accompanied by the fee set forth in § 41.20(b)(1) within the time period provided under § 1.134 of this title for reply.

Does “may appeal” refer just to the claims twice rejected, or does it refer to the whole continuation application?

I believe the application must be appealed as a whole and you designate the claims at issue in the brief. Having filed the notice without identifying the claims under appeal, in the brief can I also designate the claims that have been rejected once. Or . . . can I cancel the claim twice rejected [for another continuation] and appeal only the claims that have been first rejected? I am still one whose claim has been twice rejected. Nothing says that’s the claim I have to appeal.

In any case, if the first OA in the CIP allows some claims, you are in the same predicament as you were in originally. Appeal the lot, or split and later appeal the rejects in a continuation?

Finally, according to the statute, you can only appeal some “decision” [presumably the rejection] by a primary examiner. (The rule, however, just refers to “examiner.”) What if one of the two rejections is by a newbie examiner or by an SPE? Then the claim has not been twice rejected by a primary examiner.

Actually, . . . donkey dung is a lot clearer.