I seem to recall that there might be a USPTO rule that forbids this, but I can't seem to find it now. Am I right or wrong?
There is no such rule. Take one example - early in the process an inventor will need advice on whether or not to file, what aspects might be the most value to claim. There would be advice but not necessarily any filing. Or if a pro-se applicant was considering fully engaging a practitioner there would be advice in advance of any formal action.
The issue where this seems to be brought up are when a pro-se applicant gets some professional advice short of a registered practitioner taking responsibility via filing in the case. The (erroneous) theory is that a pro-se applicant gets some special treatment by the USPTO that would be undeserved if "secretly" getting professional advice.
Examiners are trained to consider helping pro-se applicants but that does not amount to any actual rights being accorded them, so no "rights" are being fraudulently obtained.
The USPTO's ethics rules for registered practitioners.