There are 8 symmetric Sierpinski relatives. The four videos below illustrates the construction of those symmetric relatives using just rotations (transformations 1, 2, 3, and 4) where the lower left piece is not rotated (transformation 1).
The IFS for relative 111 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.5} & 0 \\ 0 & {0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0 \\ {0.5} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.5} & 0 \\ 0 & {0.5} \\ \end{array}} \right]{\bf{x}} \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.5} & 0 \\ 0 & {0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.5} \\ 0 \\ \end{array}} \right] \\ \end{array}$$
The IFS for relative 214 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0} & -0.5 \\ 0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0.5 \\ {0.5} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.5} & 0 \\ 0 & {0.5} \\ \end{array}} \right]{\bf{x}} \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0} & 0.5 \\ -0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {0.5} \\ 0.5 \\ \end{array}} \right] \\ \end{array}$$
The IFS for relative 313 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0.5 \\ {1} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.5} & 0 \\ 0 & {0.5} \\ \end{array}} \right]{\bf{x}} \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {-0.5} & 0 \\ 0 & {-0.5} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {1} \\ 0.5 \\ \end{array}} \right] \\ \end{array}$$
The IFS for relative 412 is
$$\begin{array}{l} {f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0} & 0.5 \\ -0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} 0 \\ {1} \\ \end{array}} \right] \\ {f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0.5} & 0 \\ 0 & {0.5} \\ \end{array}} \right]{\bf{x}} \\ {f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}} {0} & -0.5 \\ 0.5 & {0} \\ \end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}} {1} \\ 0 \\ \end{array}} \right] \\ \end{array}$$
