A versor refers to a Clifford monomial (product expression) composed of invertible vectors. It is called a rotor, or spinor, if the number of vectors is even. It is called a unit versor if its magnitude is 1.
Every versor \(A=a_1 \ldots a_r, \quad a_1, \ldots , a_r \in \mathbb {R}^2, r \in \mathbb {N}\) has an inverse
\[
A^{-1}=a_r^{-1} \ldots a_1^{-1}=a_r \ldots a_1 /\left (a_1^2 \ldots a_r^2\right ),
\]
such that
\[
A A^{-1}=A^{-1} A=1 .
\]
This makes the set of all versors in \(C l(2,0)\) a group, the so called Lipschitz group with symbol \(\Gamma (2,0)\), also called Clifford group or versor group. Versor transformations apply via outermorphisms to all elements of a Clifford algebra. It is the group of all reflections and rotations of \(\mathbb {R}^2\).
The normalized subgroup of versors is called pin group
\[
\operatorname {Pin}(2,0)=\{A \in \Gamma (2,0) \mid A \widetilde {A}= \pm 1\} .
\]
In the case of \(C l(2,0)\) we have
\[
\begin {aligned}
& \operatorname {Pin}(2,0) \\
& =\left \{a \in \mathbb {R}^2 \mid a^2=1\right \} \cup \left \{A \mid A=\cos \varphi +e_{12} \sin \varphi , \varphi \in \mathbb {R}\right \} .
\end {aligned}
\]
The pin group has an even subgroup, called spin group
\[
\operatorname {Spin}(2,0)=\operatorname {Pin}(2,0) \cap C l^{+}(2,0) .
\]
The spin group has in general a spin plus subgroup
\[\operatorname {Spin}_{+}(2,0)=\{A \in \operatorname {Spin}(2,0) \mid A \widetilde {A}=+1\}.\]