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Let $\mathcal{A}$ and $\mathcal{B}$ be standard operator algebras on Banach spaces $\mathcal{X}$ and $\mathcal{Y}$, respectively. Let $\phi: \mathcal{A} \rightarrow \mathcal{B}$ be a bijective map. In this paper, we show that $\phi$ is completely preserving quadratic operator in both directions if and only if $\phi$ is 2-quadratic preserving operator in both directions and if and only if $\phi$ is either an isomorphism or (in the complex case) a conjugate isomorphism.