I recall just a few years ago, there was a common misconception that fluoride interferes with the antibacterial activity of cetylpyridinium chloride (CPC) – in fact, I've often seen instructions on antibacterial CPC rinses specifying that you should thoroughly rinse the toothpaste (which they probably assume contains fluoride) out of your mouth before using them.
Over the last few years, I've also seen a rise in alcohol-free fluoride rinses with some amount of CPC, marketed as anticavity with the "ADA Accepted" seal of approval.
However, I found this article recently: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4546200/ – a study that tested a rinse containing both common OTC mouthwash-standard concentrations of sodium fluoride and CPC – compared against fluoride-only, CPC-only, and placebo mouthwashes. In short: the study found that CPC doesn't significantly interfere with fluoride's anticavity properties, and fluoride doesn't interfere with CPC's antibacterial activity.
Additionally, fluoride and CPC both have different ways of making cariogenic bacteria miserable – for example, CPC breaks bacterial cell membranes (https://www.dentistrytoday.com/articles-hygiene/7104-its-80-20-with-biofilm-update-on-oral-rinses), while fluoride inhibits important bacterial enzymes and reduces acid tolerance in bacteria that would otherwise thrive in a low pH (https://www.ncbi.nlm.nih.gov/pubmed/7497353).
Despite all this information readily available in academic journal articles, I have yet to see a combined fluoride/CPC rinse labeled as both anticavity and antibacterial. For some unknown reason, the cetylpyridinium chloride is labeled as an "inactive ingredient". 2 examples: ACT Alcohol-Free Anticavity rinse, cinnamon flavor (https://dailymed.nlm.nih.gov/dailymed/drugInfo.cfm?setid=f2952ca5-c1c3-42fc-82a4-08662286be44) and a similar rinse by Crest (https://dailymed.nlm.nih.gov/dailymed/drugInfo.cfm?setid=52232bad-90ce-4f59-8b26-ca4c674df836)
Several years ago I recall a Listerine fluoride anticavity rinse that also contained the regular antibacterial Listerine formulation (22% alcohol and antibacterial essential oils), yet the essential oils were listed as inactive.
Why the confusion and misinformation? If a product is both anticavity and antibacterial, it should be listed properly.