OK Thomas, let's go thru some motor basics when operated on inverters.
First, the term "constant torque" or "variable torque" when applied to inverters is a totally misleading term. It has nothing to do with torque but instead refers to how much short-term overload capacity the drive has. Variable torque drives usually have 10% overload capacity and constant torque drives usually have 50%. The same drive can be rated either way simply by derating the continuous current spec for the 50% overload rating.
Secpnd, an induction motor will generally develop its nameplate rated torque at nameplate voltage and current. Use the formula hp=TxRPM/5252 to calculate torque. For example, a motor nameplated 10hp, 460V, 16amps, 1780rpm, will develop about 30ft-lbs torque when fully loaded (that is 16 amps). Since motors are primarily inductive devices, if you reduce the frequency to the motor, you will also have to reduce the voltage in the same proportion to keep the current constant. That's what an inverter does when it is set up for constant volts per hz. So, at 60hz, the motor sees 460V, at 45hz it sees 345V, at 30hz 230V, and at 15hz 115V. Under those conditions the motor will draw the same current as when loaded at nameplate conditions. This means that an induction motor can be expected to produce constant torque from its nameplate base speed down to near zero speed if the V/hz ratio is held constant. Referring to the formula above, you can see that the hp drops linearly with speed from 10hp at nameplate speed down to zero at zero speed.
Third, when an induction motor is driven into its overspeed range by exciting it with a frequency over its nameplate, clearly you cannot continue to increase the voltage above the motor rating. So, above base speed, the voltage is held constant and the current that results is also constant. The motor input kw is therefore constant. It should come as no surprise then that the motor output kw or hp should also stay constant. Looking at the above hp formula again, you can see that, for hp to hold constant with increasing rpm, the torque must drop down in the same proportion as the rpm is increased. So, above nameplate base speed, the motor produces constant hp which results in the torque dropping as you go further and further into overspeed. This cannot go on indefinitely without the motor flying apart or, which usually happens first, the motor windings begin to misbehave due to the high frequencies and the motor hp actually starts to drop too. This varies some with different motor designs but, under 150hp, you can expect constant hp out to at least 90hz.
When applying an overspeed motor to a load, you must be sure that the load torque is always less than or equal to the available motor torque or you will not be able to drive thru that speed where the motor torque is less than load torque. If there is reduction in the power train, you must factor that into the calculation of torque too.
In summary, generally you can expect an induction motor on an inverter to develop constant torque at and below base speed and constant hp above base speed. All of this within reasonable limits.