OK, Tim, I'll try to lay out some of the technical aspects of three phase induction motors in ordinary terms in the best way that I can.
First, I assume that you have examined a NEMA Design B torque-speed curve. It shows the motor shaft speed at no load being right at the synchronous rotation speed of the stator's magnetic field at 60Hz (ie. 3600,1800,1200,or 900rpm depending upon the number of magnetic poles wound into the stator coils. Then, as some load begins drawing torque out of the motor rotor, the rotor begins to slow down slightly (this is called "slip") until, at the nameplate rated torque level, the rotor has slowed to the nameplate speed. For example, in a common 4 pole motor at 60hz the stator field is spinning at 1800rpm and at no load the rotor is alsos spinning very close to 1800rpm as well. If the motor is loaded up to its nameplate output (let's just say 10hp), the shaft speed will drop to the nameplate speed (let's say 1760rpm). That 40rpm slip (1800-1760=40) is directly proportional to shaft torque. So, at 20rpm slip you would be at 1/2 torque, at 30rpm it would be 3/4 torque, and so on. Since HP=T x rpm/5250, in the above 10hp motor the full load torque would be about 30ft-lbs. You will also notice that, on the NEMA B curve, the motor can actually deliver up to 220% of nameplate torque before it begins to enter the breakdown region where current continues upward but torque fall off. This is due to the magnetic circuit of the motor beginning to saturate. When that happens, magnetic chaos ensues, the motor becomes much less efficient, and, if left to operate in this region would overheat rapidly and destroy itself.
It is important to see that the NEMA B curve and everything I've mentioned above about it pertains to a power supply of 60hz and proper nameplate voltage.
If the frequency of the incoming power is reduced, the stator field begins to spin slower and the resulting speed of the motor shaft is similarly reduced. In fact, the motor's ability to make torque in its shaft is dependent upon the ratio of the voltage to the frequency. For example, a 480V 60hz motor needs a ratio of 8/1 to produce nameplate torque. A 240V 60hz motor has a 4/1 ratio and a 400V 50hz European motor has an 8/1 ratio.
If we reduce the power frequency fed to a motor and simultaneously reduce the voltage to maintain the same ratio (480/60, 400/50,240/30,120/15 etc), we can get the motor to slow down and continue to produce the same torque as long as we maintain the same volts per hz ratio. This is exactly how an inverter gets a motor to change speed.
From the above, it should be clear that, as long as we hold the v/hz ratio constant, the motor will be able to source constant torque levels as we slow down. This is actually the case down to about 1.5hz where the motor becomes impossible to control using simple v/hz speed control.
So, we have a motor (in fact, any induction motor) that can source constant torque down to nearly zero speed as long as we hold the v/hz ratio constant. The problem with this is that, while magnetically able to do this, the motor is NOT thermally able to do it. As the motor slows down, the shaft mounted cooling fan (internal in ODP motors and external in TEFC motors) also slows down and the motor's ability to cool itself goes away. So, it can be said that all induction motors are capable of constant torque output over the speed range of near zero up to motor nameplate or base speed magnetically but not thermally. Most motor manufacturers will list the "speed turndown" ratio of their motors which would typically be 4/1 Constant Torque. This means that the cooling system designed into the motor will keep the motor within its design temperature limit from 60hz down to 15hz (4/1) while working over the speed range at full torque. Sometimes you will see motors listed at 10/1 or even 1000/1. Motors with very high turndown ratios will generally not have a shaft fan but some other method of cooling. Auxiliary fans which have their own power source and run at constant speed regardless of motor shaft speed or TENV (Totally Enclosed Non-Vented) motors would be like that.
As you mention, Tim, sometimes you will see motor speed turndown ratios listed as, for example, 4/1 CT, 10/1 VT. The CT is Constant Torque loading as described above but the VT is variable torque and clearly has a wider speed turndown range. The difference is in the load connected to the motor, not the motor itself.
A variable-torque load is normally a load like a fan or centrifugal pump that does not draw the same amount of torque at all speeds. Typically, the torque begins at essentially zero at zero speed and increases by the square of the speed up to the design load and speed.