Tail-biting trellises and their pseudocodewords are very important for modern decoding techniques like iterative decoding. We introduce a useful matrix representation of trellises, give its fundamental properties, and use it to enumerate and describe the distribution of trellis pseudocodewords. We give several examples, a couple of which lead to important open problems. Next, we prove that the pseudocodeword weight enumerator introduced in [2] always satisfies a recurrence equation, and, for certain trellises including the Golay trellis given in [3], that it is invariant under generalized MacWilliams transformations, allowing invariant theory to be used for computing it. Computation for the Golay trellis shows then that pseudocodewords of period at most 4 must have AWGN pseudoweight at least 8.