I think you've perhaps misunderstood the concept of exhaust scavenging slightly in asking if burned gasses would be mixing with fresh charge, especially on a muli-piston engine.
Bear in mind that the design of the rotating assembly is such that the piston will move at it's fastest vertically when mid-bore. Once it approaches TDC it's height relative to the deck will be pretty constant within a small percentage for a good percentage of crank rotation.
It is at this point where valve overlap plays it's part but the piston has already pretty much pushed the entirety of the exhaust gas out of the manifold. What has effectively happened is that a "pulse" of pressurised gas has been sent down the manifold but at the point where the primaries meet this will cause a pressure differential between exhaust headers.
At the point where the inlet valve opens, the piston crown is already very close to TDC where it will say for a fair duration as the crank rotates into position ready for the inlet stroke. The effective negative pressure at the exhaust valve caused by the pressure waves now exiting the exhaust tailpipe and by the actions of other phased pulses in other primaries means that the exhaust valve is acting as a vacuum, acting in the bowl of the piston and the squish band within the head (depending on design).
What happens when the inlet valve opens is that this vacuum is drawing fresh mixture from the inlet manifold. If the overlap is too long it's more likely that unburned fuel will enter the exhaust system than burned gas re-enters the induction chamber.
I'm sure you can imagine how the inlet and exhaust valves on a reverse port engine would therefore create swirl on the piston crown. Although in most implementations this is limited as reverse port engines are typically 1 inlet and 1 exhaust valve per cylinder which are usually in a line above the centre of the combustion chamber.
There is a further limitation to consider with real-world reverse port engines and that is the physical space on the manifold side of the cylinder head. Austin-Morris on the A series Mini engine got around this to some extent by using siamese ports but this isn't ideal as it includes it's own challenges. Volkswagen, on the original Golf GTI DX engine, mounted the inlet manifold slightly above the exhaust but this leads to heat soaking up from the exhaust manifold.
Cross-flow designs allow much larger ports as you have a bigger surface are in which to cast your inlet and exhaust ports. It also allows you to fit multiple valves and twin-cam cross-flow designs allow inlet and exhaust valves to be placed either side of the centre-line which again will promote swirl. It will also allow central placement of the spark plug. In real-world flow terms, a cross-flow cylinder head has many advantages over reverse flow.
There is of course a third way...
The biggest problem with either cylinder head design is that the inlet temperatures are typically way lower than the exhaust temperatures. This leads to a "hot side" of the cylinder head which is ultimately an engineering limitation in terms of peak power.
Lancia, in the 1980's, pioneered a design they named triflux. The inlet manifold was mounted centrally between both cams. There was then an exhaust manifold on each side of the cylinder head. Each cylinder had two inlet and two exhaust valves but they were alternated; like the criss-cross lacing of running shoes. This allowed for excellent swirl and scavenging but prevented there from being a "hot side" of the engine. With the engine heating up uniformly it allowed far greater power output. Indeed they achieved over 900bhp from a 1.7 litre inline 4-cyl using a twin-turbo implementation of this design.
Ultimately, for road cars, there is always a trade-off between power, efficiency and tuning capability and cost of manufacture. In terms of engines you are likely to encounter as opposed to design from scratch, the cross-flow offers a better package in terms of flow simply because it has twice the surface area on the cylinder head upon which you can mount your manifolds.