I'd assume that portals provide a connection between two points in space, and not a connection between parallel universes, thus there is still only one rod of the same mass even when it is adhered. If you take the rod and divide it up into small elements, each element is connected to the rod above it and below it. Thus to draw a free body diagram of the element, there would be gravity acting downwards and equal tensile forces acting both downward and upward. Since the tensile forces are equal, they cancel out, thus the only force acting on the element of the rod is gravity, causing it to accelerate down. Because the rod has no end (they were adhered together), all elements in the rod act the same way, thus the entire rod accelerates down at g, the acceleration due to gravity. Of course, assuming this isn't done in a vaccuum, there will be drag associated with air viscosity which will eventually limit the velocity of the rod to what we call its terminal velocity.
If one portal instantly changes places to a point farther away, we end up with a complete rod of the original length. There is still an adhesion somewhere along its length, but when one portal was switched, it severed the connection between the two ends currently intersecting the portals, giving the bar back its ends.
Thus if the portal was moving in small increments, we would observe a bunch of chopped slices of the rod. If it was moving smoothly and continuously, we would see a fine dust of shavings of the rod.