Why is there an asymmetry between the left and right annihilators?

Question

Let $X$ be a normed vector space, and $X'$ be the dual space of $X$, consisting of all bounded linear functionals on $X$. If $K\subset X$ and $L\subset X'$, then we may define the left and right annihilators to be: $$^\perp L=\{x\in X:\langle x',x'\rangle=0,\forall x'\in L\}$$ $$K^\perp=\{x'\in X':\langle x',x\rangle=0,\forall x\in K\}$$ So $^\perp L$ should be the set of all vectors annihilated by every functional in $L$, and $K^\perp$ should be the set of all functionals that annihilate every vector in $K$. These two concepts seems somewhat symmetric, however, we always have: $$^\perp(K^\perp)=\overline{\text{span}(K)}$$ But only: $$(^\perp L)^\perp\subset\overline{\text{span}(L)}$$ where in the second statement, the reverse equality may sometimes fail. I think this is really weird, because these two seemingly symmetric concepts are actually asymmetric. Can someone explain why that happens to me? Does it have something to do with the complete nature of $X'$, despite whether $X$ is complete or not? Would these two be symmetric, if we additionally assume $X$ is complete? Or assume that $X$ is reflexive?