The 2D rigid body transformation matrix I'm concerned with is in homogeneous representation, hence the form:
\begin{bmatrix} \textbf{R} & \textbf{t} \\ 0^T & 1 \end{bmatrix}
I know the orthonormal constraints: $$R^TR=1,$$ which contributes 3 constraints in the 2D case.
But does the homogeneous representation itself provide some constraints? And how many constraints does the translation vector $t$ provide? Thus, the overall number of constraints is 6 in 2D case.
The group of rigid transformations is called the special euclidean group and denoted by $\textrm{SE}(2)$ in 2D and $\textrm{SE}(3)$ in 3D. This group is well known and is defined as follows: $$ \textrm{SE}(2) := \biggr\{ \mathbf{T} = \begin{bmatrix} \mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix} : \mathbf{R} \in \mathrm{SO}(2), \mathbf{t}\in \mathbb{R}^2 \biggr\}$$ $$ \textrm{SE}(3) := \biggr\{ \mathbf{T} = \begin{bmatrix} \mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix} : \mathbf{R} \in \mathrm{SO}(3), \mathbf{t}\in \mathbb{R}^3 \biggr\}$$ where the group of rotations is called the special orthogonal group and denoted by $\mathrm{SO}(2)$ in 2D and $\mathrm{SO}(3)$ in 3D. The special orthogonal groups are defined as $$ \textrm{SO}(2) := \bigr\{ \mathbf{R} \in \mathbb{R}^{2 \times 2} : \mathbf{R}\mathbf{R}^T = \mathbf{1}, \det{\mathbf{R}} = +1 \bigr\}$$ $$ \textrm{SO}(3) := \bigr\{ \mathbf{R} \in \mathbb{R}^{3 \times 3} : \mathbf{R}\mathbf{R}^T = \mathbf{1}, \det{\mathbf{R}} = +1 \bigr\}$$ where the determinant is positive because the group does not include reflective transformation. If we allow $\mathbf{R} = \pm 1$, then reflection are allowed, but I do not know if there is a name for this group.
To answer your question, no additional constraints exist beyond what is defined. Adding additional constraints would make a subgroup.