This is the problem…I don’t believe relays are considered SPDT or SPST. A relay normally has 3 connections, the com or common and the NC and NO which is Normally Open and Normally Closed. Like the name sounds, when power is NOT applied to the relay, it connects the NC contact to common and when it is applied it switches to the NO contact to common, disconnecting the NC. That’s a relay circuit. What you are talking about is a specific type of switch that has 3 connections, so that you can switch between A or B. This is the best diagram I could find quickly.
This switch only has 2 positions, A or B, but instead of just cutting A when you flip the switch, it also connects B. That a SPDT. You can also have a SPDT switch like @JDRoberts described, which is an On off On SPDT switch that has 3 positions. Position A would connect one side of the switch, the middle would connect neither and Position B would connect to the other side of the switch. The bridge that @ehupp01 posted would be a single relay.
So, what you really need to describe to us, or find online, is what the current switch you have is actually doing electrically. When you flip the switch does it remain latched or is it momentary? Is it an On off On SPDT switch? If so, I can’t think of anything that would accomplish that. What you could do, and this is ONLY if the switch is actually a SPDT On-On switch is to connect the down position to NC on the relay and the up position to NO. This would mean that when the relay is powered off, it would go down and when you turned the relay on it would go up. But, that is ONLY if it is latched and On-On. If it is momentary, your job becomes a LOT easier.
If you should wire it up the way I said and the current switch is something other than an ON-ON SPDT switch, its possible that wiring it this way could burn out the motor, the control board or both or start a fire. So…you want to be 199% sure before you start monkeying with it that you know how the current switch works. I would suggest some time with a multimeter would be in order.