In Higson and Roe's Analytic K-homology, for a unital $C*$-algebra $A$, the definitions of K-theory and K-homology have quite a similar flavor.

Roughly, the group $K_0(A)$ is given by the Grothendieck group of homotopy classes of matrix algebra projections over $A$.

On the other hand, $K^0(A)$ is the Grothendieck group of homotopy classes of even Fredholm modules over $A$ (or more correctly unitary classes of Fredholm modules).

Now in contrast to the $K_0(A)$ case, every element of $K^0(A)$ contains a representative Fredholm module. (For the inverse of the class of $(H,\rho,F)$, just take the class of $(H^{\text{op}},\rho,-F)$, where op denotes the opposite grading.

Does this means that taking the Grothendieck is not strictly necessarily to define $K^0(A)$? Could one just as well define it as the monoid of homotopy classes of Fredholm modules, and then prove that it was a group?

If this is true, then is there any deeper philosophical reason why $K_0$ requires us to introduce inverses, while $K^0$ does not?