Intersections of some families of languages

What do a pushdown and a queue have in common? What is their intersection? Is it a counter? If we add the one-reversal restriction, is a one-reversal counter exactly the intersection of a one-reversal pushdown and a queue or, symmetrically, the intersection of a one-reset tape and a pushdown? These and similar assumptions can be heard here and there, and there are some conjectures by Autebert et al. [1], Book et al. [3] and RodriguezWe disprove all these conjectures and show that counters are strictly weaker than the intersection of pushdowns and queues. This goes through for the restriction to one reversal or one reset. In fact, we obtain new families of languages from intersections of some well-known families.