We make use of the fact that a two-sided whole-plane Schramm-Loewner evolution (SLE$_κ$) curve $γ$ for $κ\in(0,8)$ from $\infty$ to $\infty$ through $0$ may be parametrized by its $d$-dimensional Minkowski content, where $d=1+\fracκ8$, and become a self-similar process of index $\frac 1d$ with stationary increments. We prove that such $γ$ is locally $α$-Hölder continuous for any $α<\frac 1d$. In the case $κ\in(0,4]$, we show that $γ$ is not locally $\frac 1d$-Hölder continuous. We also prove that, for any deterministic closed set $A\subset \mathbb{R}$, the Hausdorff dimension of $γ(A)$ almost surely equals $d$ times the Hausdorff dimension of $A$.