Geometric understanding of why $D_1\cong \mathbb{Z}_2$

Question

From what I understand, $P_2$ is the line joining $(-1,0)$ to $(1,0)$ and then $P_1$ is the point $(1,0)$. This is due to defining $P_n$ (the regular polygon with # of points $n$) to be formed by the successive joining of the points: $$\begin{pmatrix} \cos \left ( \frac{2\pi k}{n}\right ) \\ \sin \left ( \frac{2\pi k}{n}\right ) \end{pmatrix}, \; k=0,1,2,..., n-1$$ Now my problem is that for any point, I'm understanding that the symmetry transformations (for a single point) consists of all rotations and reflection with respect to that point. Then how can this dihedral group for $n=1$ only contain two elements? Namely the reflection along the $x$ axis and the identity. I know that the order of $D_n$ is $2n$ however I'm just struggling to understand this for $n=1$.