4 Pin Potentiometer Identification
I second Spehro in what he thinks it is: an heavy-duty pot with additional center tap.
TT Electronics seems to sell a product similar to the part sported in your photo (datasheet).
Here is an excerpt from the datasheet with the technical drawing:
And here is the electrical circuit:
Even if it is not what you are looking for, it may be a starting point.
Not familiar with that series, but it appears to be an extended-life pot.
There are two possibilities for the 4th connection. The first is that it is a center-tap on the element. This might allow them to control the output at the neutral position of the joystick better. The second (and less likely one) is that it's a ground connection.
A quick test with an ohmmeter should settle the matter. Here is a picture of a center-tapped 10K pot. It would measure 5K from either end of the element to the opposite (center-tap) pin. The wiper would short to the center tap when in the center of rotation.
Four-terminal potentiometers were common at one stage on hi-fi volume controls to give automatic "loudness" control - see What's a potentiometer with four terminals in this schematic? - but these were logrithmic law and a joystick is more likely to be linear.
My guess is that it's a linear pot and that the centre-tap is used to provide a definite GND or reference to assist with a deadzone in the centre of the pot. This would be intended to prevent creep when the joystick is centred.
simulate this circuit – Schematic created using CircuitLab
I found a Clarostat brochure on Stephen Engineering and they seem to be handling the product now. They don't have any four terminal pots listed.