Is it possible to recover the seed value from a Mersenne Twister PRNG state? So I know that you can recover the state of a mersenne twister generator by observing a sequence which is long enough. 

I was wondering if you could be able to recover the actual seed value used or if an attempt would yield an equivalent at best?

Also, closely related, let's say we use a 32-bit seed value. Does this mean that a mersenne twister generator could yield $2^{32}$ different sequences for each unique seed value or are there seeds which produce the same sequence?
I was wondering if you could be able to recover the actual seed value used or if an attempt would yield an equivalent at best?


Yes (and it turns out to be not that expensive computationally).

The Mersenne twister state is an invertible linear function of the seed, hence recovering the unique seed is possible.


  Also, closely related, let's say we use a 32-bit seed value. Does this mean that a mersenne twister generator could yield $2^{32}$ different sequences for each unique seed value or are there seeds which produce the same sequence?


Every seed will generate a unique sequence.  This is a stronger statement than what I just previously made (it would be conceivable that two different states would generate the same sequence - however, that doesn't happen with Mersenne).