Turning an Internal Taper

SNNC 156 P1 Internal Taper Turning
Jack is making a right angle adaptor for his milling machine. It's quite a project involving turning pieces on a lathe, doing some milling on his vertical milling machine and a bit of welding. One of the last bits is to cut a tapered hole in the output shaft of the adaptor to accommodate a collet to hold the bit or tool. Looking for advice, he turns up this YouTube video about turning an Internal Taper.

When you get to the part where he is actually turning the taper (around about the 7 minute mark), Jack noticed that the cutting tool is upside down and wonders why. This is one of those situations where you have multiple binary choices. Inside or outside? Clockwise or counter-clockwise? Taper front-to-back or back-to-front? Now, which side do you cut on, and which way should the cutter face? It is like a computer programming problem, where you have multiple overlapping choices and you end up constructing a five line 'if' statement to cover all the conditions. I started writing a description of why and wherefore and decided if I was going to help, a picture would be better, so I sketched one.

Lathe arrangement for cutting a taper.

The language in the video might be English, but if it is, it is so heavily accented that I find it almost unintelligible. Jack claims to have no problem understanding him. I can pick out one word out of ten.