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A version of the classical Klee-And\^o Theorem states the following: Forevery Banach space $X$, ordered by a closed generating cone $C\subseteq X$,there exists some $\alpha>0$ so that, for every $x\in X$, there exist$x^{\pm}\in C$ so that $x=x^{+}-x^{-}$ and$\|x^{+}\|+\|x^{-}\|\leq\alpha\|x\|$.  The conclusion of the Klee-And\^o Theorem is what is known as a conormalityproperty.  We prove stronger and somewhat more general versions of the Klee-And\^oTheorem for both conormality and coadditivity (a property that is intimatelyrelated to conormality). A corollary to our result shows that the functions$x\mapsto x^{\pm}$, as above, may be chosen to be bounded, continuous, andpositively homogeneous, with a similar conclusion yielded for coadditivity.Furthermore, we show that the Klee-And\^o Theorem generalizes beyond orderedBanach spaces to Banach spaces endowed with arbitrary collections of cones.Proofs of our Klee-And\^o Theorems are achieved through an Open Mapping Theoremfor cone-valued multi-functions/correspondences.  We very briefly discuss a potential further strengthening of The Klee-And\^oTheorem beyond what is proven in this paper, and motivate a conjecture thatthere exists a Banach space $X$, ordered by a closed generating cone$C\subseteq X$, for which there exist no Lipschitz functions$(\cdot)^{\pm}:X\to C$ satisfying $x=x^{+}-x^{-}$ for all $x\in X$.

Strong Klee-And\^o Theorems through an Open Mapping Theorem for cone-valued multi-functions