Can a computational complexity class be redefined by using any complete (decision) problem?
If $X$ is any NP-complete decision problem, then NP consists of all decision problems which are polytime many-one reducible to $X$.
We cannot replace polytime many-one reductions with polytime Turing reductions, however (unless NP=coNP), since coSAT is polytime Turing reducible to SAT.