Jelly, 12 bytes
Employs the inverse test of a post by pxeger - here.
JŒ!<ƝṢƑ$Ƈịċ⁸ A dyadic Link that accepts the sign, \$S\$, on the left and the deck of letters, \$\sigma\$, on the right and yields a positive integer (truthy) if possible or \$0\$ (falsey) if not.
Try it online! Or see the test-suite.
How?
First, this constructs a list of all permutations of the indices of \$S\$ which do not decrease after their first increase.
e.g. for \$S\$ of length four:
[1, 2, 3, 4] [2, 1, 3, 4] [3, 1, 2, 4] [3, 2, 1, 4] [4, 1, 2, 3] [4, 2, 1, 3] [4, 3, 1, 2] [4, 3, 2, 1] It then uses these to index into the deck, \$\sigma\$, to construct all the possible signs and counts the occurrences of the sign, \$S\$.
JŒ!<ƝṢƑ$Ƈịċ⁸ - Link: list of characters S, list of characters D (sigma) J - range of length (S) -> [1,2,3,...,length(S)] Œ! - all permutations Ƈ - filter keep those for which: $ - last two links as a monad: Ɲ - for neighbouring pairs: < - less than? Ƒ - is invariant under?: Ṣ - sort ị - index into (D) (vectorises) ⁸ - chain's left argument = S ċ - count occurrences