Is there any advantage, disadvantage, safety concerns, etc, in twisting, or not twisting stranded wire, such as fixture wire, to the solid wire before installing the wire nut?

I always twist solid wire with linesman pliers, then install the wire nut. With fixture wire, I just twist the fixture wire by hand and install the wire nut. I’ve heard reasons for doing, or not doing both (some sound really unbelievable) and was wondering if there are any valid reasons to this, or are these just “preferences” that have grown legs over the years.

When you distill this information:
Wire nut - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Wire_nut)

What I think I find is that early wire nuts without the spring, REQUIRED that the wires be twisted together before the wirenut is installed.

With the spring version as the reference suggests, the spring CUTS into the wires and twisting is not required.

So, if anything, the twisting is a carryover. With modern nuts, especially the ones with wings and a spring it's not necessary.

What's more important to to the connection is the proper strip and the number and size of the conductors.

Read and follow the manufacturer's instructions for each product you use.