There are ten
possible symmetries using polyhex pieces. I use the
convention that the pieces are always shown with a flat side
at the top and bottom, and points at the sides. The terms
"Horizontal" and "Vertical" apply to patterns using this
convention. 

No. 
Type of Symmetry 
Example 
Notes 
1 
Full 

The maximum number of symmetries for
the polyhexes is 12. There are six rotations and six
reflections. A solution must
contain n x 6 or n x 6 + 1 unit
hexes. 
2 
Sextarotary 

A pattern with six rotations. A solution must contain n x 6 or n x 6 + 1 unit hexes. 
3 
Ternary & Horizontal 

A pattern with three rotations and three reflections, with the mirrors through the faces of the central hex. A solution must contain n x 3 or n x 3 + 1 unit hexes. 
4 
Ternary & Vertical 

A pattern with three rotations and three reflections, with the mirrors through the points of the central hex. A solution must contain n x 3 or n x 3 + 1 unit hexes. 
5 
Trirotary 

A pattern with three rotations. A solution must contain n x 3 or n x 3 + 1 unit hexes. 
6 
Double Bilateral 

A pattern with two perpendicular reflections. A solution may contain any number of unit hexes. 
7 
Birotary 

A pattern with two rotations. A solution may contain any number of unit hexes. 
8 
Horizontal 

A pattern with one reflection, with
the mirror through the faces of the central column. A solution may contain any number of unit hexes. 
9 
Vertical 

A pattern with one reflection, with
the mirror through the points of the central row. A solution may contain any number of unit hexes. 
10 
None 

A pattern with no symmetries. A solution may contain any number of unit hexes. 