I have been sampling the last few days...tailspinning with and without cores, and corespinning clockwise or counterclockwise. Above is tail spinning of lovely mohair without a core. The yarn is a single, sturdy and relaxed. I thought it would be either too tight or fall apart. I did have drift apart with some slippery joins that did not survive the skeining...but the small skein that remained is very nice!
The 3 photos above are tail spun locks, spun clockwise over a handspun single, thus tightening the single. This was easy to do and surprisingly did not make a wirey nor an unduly overspun single. It's not like you are likely to knit a tailored garment with tailspun, so biasing is not likely to be a problem.
The tailspun in the 2 photos above was spun counterclockwise over a handspun single, thus loosening the twist of the core. It was a bit harder to do, but the yarn is nice and soft and sturdy. Not a whole lot different than the larger hank spun in the opposite direction.
The final samples were true corespun. I handcarded tiny batts of mohair and spun them horizontally over a commercial core: cheap pink acrylic yarn from the thrift store. In this test I saw a very big difference between spinning clockwise and counterclockwise. The acrylic yarn tightened dramatically when I spun counterclockwise, making a tight, overspun hank. The core that was spun clockwise made a lovely, glossy, relaxed single with only a touch of bias.
Moral of the story? Pay attention to the direction that your core was originally spun in when doing classical core spinning. When spinning tails? The jury is out. I like the tail spun on a core somewhat better than the core-less yarn...but the affect of direction on the twist of the finished yarn was harder to discern.