Does the rotary trumpet have a cylindrical bore like the piston trumpet, or a conical bore like the cornet? I would assume that it's cylindrical, but for the fact that it has a mellower tone and seems to be less prone to becoming overly brassy.
All valved cornets and trumpets have a mouthpipe whose bore diameter tapers (gradually increases in internal diameter) from the mouthpiece receiver until:
either the mouthpipe joins a tuning slide. By far the most common arrangement is that once the mouthpipe joins the tuning slide the bore then stays cylindrical until it emerges again into the bellpipe. Some manufacturers do offer a dual-bore tuning slide, where the two legs of the slide have different bores but this is very unusual.
or the mouthpipe joins a valve. Some higher pitched trumpets and cornets have a moveable leadpipe rather than a tuning slide, and this allows the mouthpipe to taper all the way to the valve.
In trumpets the bell pipe emerging from the valve block stays cylindrical for quite a while before it starts to flare, whilst a cornet bell pipe starts to flare immediately.
And the bell curvature in the final 15cm or so of trumpets and cornets differs too - trumpet bells tend to flare later, and sound brighter. Cornet bells start to widen earlier, they have less severe curvature and sound slightly warmer.
Piston valve trumpets and cornets (lower pitched ones at least) have the lead pipe entering the third valve, emerging from the first. This allows the lead pipe to be relatively long, giving the maker more opportunities to modify the taper to improve the playing characteristics.
Conventional rotary-valve trumpets and piston-valve flugel horns have a short tapered leadpipe into the first valve - so the bore must become cylindrical at that point. Once it emerges from the third valve it then remains cylindrical for quite a while (look at some pictures) before it starts to flare. The flugel horn, noted for its warm sound also has a short leadpipe into the first valve, but like the cornet starts to flare almost immediately it emerges from the third valve.
To summarize: all trumpets and cornets are cylindrical in part and conical in part. What gives them their individual sound is whereabouts in the length of tubing the cylindrical part occurs, and the rate of change of the taper for those non-cylindrical parts.
The cornet is largely conical, the trumpet largely cylindrical. Neither are exclusively either.
You might find the following opinion, from a dealer in trumpets, interesting. Current German rotary trumpets appear to be both smaller bore and less conical than the familiar American piston trumpets.
Austro-German rotary valve trumpets have in general a much smaller bore than the piston valve instruments commonly in use in the United States. This does not, however, mean that rotary valve trumpets are necessarily less free-blowing. We suspect that many American orchestral players have had experience only with poorly designed or produced instruments which were stuffy or unresponsive. Bach-Selmer B♭ piston valve trumpets have bore sizes as follows: medium bore, .453", medium large bore, .459", large bore, .462"; most other American-style piston valve trumpets have similar bore sizes. By comparison, our Dotzauer rotary valve trumpets in B♭ have a bore size of .429", which is much smaller. This does not mean, however, that these instruments are small in tone or volume; a comparison of bore size alone is virtually meaningless, since it is only one factor in the overall acoustical design of the instrument. For example, rotary valve trumpets have much larger and heavier bells, have less conical and therefore more cylindrical tuning, have a larger bore mouthpipe, and are often made of gold brass rather than the yellow brass used for most piston trumpets.
You can find examples of cornets that have more cylindrical portions than some trumpets. There is no hard and fast rule any more, because the two instruments were at one point in time more different than they are now.
Cornets got a bit trumpet-ey and trumpets got a bit cornet-ey in the 20th century, especially the first half. Cornets DO start out smaller than trumpets, that's currently about the only hard and fast rule. Otherwise, you name the rule and you can likely find at least one exception.
The shapes and scales of the tapers involved do vary considerably among the various instruments, and that does affect how they sound. (How small does it start out? How fast does it get to how much bigger? Are there tapers in the crooks, like asymmetric tuning slides? How/when does the bell taper?)