When people assert implications, they often implicitly involve universal quantification. For example, "if $n$ is a prime number greater than $2$, then $n$ is odd" really means "for all integers $n$, if $\dots$." When one denies an implication, one includes the universal quantifier in the denial, so it becomes an existential quantifier. For example, if someone says that "$n$ is odd" doesn't imply "$n$ is a prime number greater than $2$", he normally means to deny that "for all $n$, if $n$ is odd then $n$ is a prime number greater than $2$"; equivalently, he means to assert that "there exists an odd $n$ that is not a prime number greater than $2$." (Recall from propositional logic that the negation of $B\implies A$ is equivalent to $B\land\neg A$.) So your proposed combined connective, for implication in one direction and denial of implication in the other direction, will implicitly quantify the variables partly with universal quantifiers and partly with existential ones. This looks to me like a recipe for confusion and therefore well worth avoiding.
If, by good fortune, your statements $A$ and $B$ don't involve variables, so these quantification issues don't arise, then there is a fairly easy answer to your question. As I said above, the negation of $B\implies A$ is equivalent to $B\land\neg A$. Furthermore, this formula already implies that $A\implies B$, so
(A\implies B)\land\neg(B\implies A)
is equivalent to $B\land\neg A$. But remember, this use of propositional logic is legitimate only if your $A$ and $B$ don't involve any variables that are implicitly quantified in your implications.