Would setting a magnetic base to the “off” position remove the torque from an external field?

A magnetic base is a permanent magnet mostly inside a ferromagnetic case. It can be rotated so that the field is either internally shunted through the ferromagnetic blocks, or so that the blocks are excited and polarized, producing a strong external magnetic field that can hold strongly to an external ferromagnetic surface like a steel bench or housing of a piece of equipment.

If it is set to "on" and not stuck to anything, it will have a strong external dipole moment, and I believe it will experience a substantial torque if there is an external uniform magnetic field.

Question: If it is set to "off" when there is minimal external field, would this substantially minimize the torque from an external uniform field? What is the underlying physics here - do I need to carefully treat the field inside the steel and permanent magnet, or just the part of the field "outside" the device to understand the main effect of "on" vs "off" on the possible torque?

Because even the torque on a magnet can usually be nulled at certain angles, assume this is repeated at a variety of angles.

I'm using the magnetic base as a real-world example, but my question is really about a macroscopic block of permanent magnet magnetized uniformly, and any kind of ferromagnetic material - used vs not used - as a "keeper" or shunt. Assume the material is linear and very high permeability to keep the problem simple.

below: an animation of a Magnetic Base (rotated) from here and here. When the indicator points to "Off", the field (unfortunately not shown) leaves one pole, splits left/right and each half is shunted in-the-plane through the steel, and emerges near the other pole. But when the indicator points to "On", The left piece of steel become a "North pole", and the right becomes a "South pole" as seen externally.

below: an example of a Magnetic Base (rotated) from here.

• Is it shielded? One can easily check with a magnetometer. There is one in every smart phone these days. Brave new world... where half the world has one of those in their pockets. :-) – CuriousOne Aug 4 '16 at 6:02