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Let $𝒩=N_n\left(R\right)$ be the algebra of all $n\times n$ strictly upper triangular matrices over a unital commutative ring $R$. A map $\varphi$ on $𝒩$ is called preserving commutativity in both directions if $xy=yx\iff \varphi \left(x\right)\varphi \left(y\right)=\varphi \left(y\right)\varphi \left(x\right)$. In this paper, we prove that each invertible linear map on $𝒩$ preserving commutativity in both directions is exactly a quasi-automorphism of $𝒩$, and a quasi-automorphism of $𝒩$ can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.