Is the SK2 calculus a complete basis, where K2 is the flipped K combinator?
Consider the terms of the $S,K_2,I$ calculus as trees (with binary nodes representing applications, and $S, K_2$ leaves representing the combinators.

For example, the term $S(SS)K_2$ would be represented by the tree

        @
       / \
      /   \
     @    K2
    / \
   /   \
  S     @
       / \
      /   \
     S     S


To each tree $T$ associate its rightmost leaf, the one you get by taking the right branch at each @. For example, the rightmost leaf of the tree above is $K_2$.

As can be seen from the ASCII art below, all reduction rules in the $S, K_2, I$ calculus preserve the rightmost leaf.

         @                           @
        / \                         / \
       /   \                       /   \
      @     g    [reduces to]     @     @
     / \                         / \   / \
    /   \                       e   g f   g
   @     f                 
  / \
 /   \
S     e


      @
     / \
    /   \
   @     f    [reduces to]   f
  / \
 /   \
K2    e


From there on, it's easy to see that if some term $T$ reduces to $T'$, then $T$ and $T'$ have the same rightmost leaf. Hence, there is no term $T$ in the $S, K_2, I$ calculus such that $TK_2S$ reduces to $K_2$. However, $KK_2S$ reduces to $K_2$, hence $K$ cannot be expressed in the $S,K_2, I$ calculus.