To cancel it down, there are only three pieces that have at least one line of symmetry: The I piece, the square and the T piece. I'll start off with the I piece because it has properties that are a bit interesting and relate to the other two.
The I Piece
This piece is probably the most important piece in the game, as general strategy revolves almost completely around it. The most obvious reason is that the I piece is the only piece in the game that can be used to clear four lines, so is therefore very valuable in relation to increasing score. As a result, this means that the best way to achieve a high score is to keep one column open whilst building with pieces on the rest of the board, and then drop an I piece down the gap for a tetris once the surrounding columns are high enough.
Strangely, there is actually an optimal column to leave open, due to the properties of the board and the I piece. The board is 11 blocks wide, and the I piece is 4 blocks wide when positioned horizontally. Consequently this means that the I piece must favour one side of its width as an axis to rotate on, rather than rotating from its middle point, due to the restrictions of the odd number of block widths that comprise the board. In the case of the NES version, the I piece favours a right side rotation, meaning that it has more space to rotate on the right side of the screen. It is therefore a better strategy to leave the far right column clear for the I piece than the left, and in some cases it may be preferable to leave a middle column open rather than a left column. Interestingly, the opposite is true on the Gameboy version.
To summarise, the I piece is both horizontally and vertically symmetrical, but it can never be placed symmetrically in relation to the board's centre or rotated on it's central axis due to the width of the board.
The T Piece
The T piece can rotate along it's exact central axis, and is the only symmetrical piece in the game that can be placed in the centre of the board. The rotational properties of the T piece give it a lot of flexibility, enabling it to be tucked into various gaps that other pieces can not fill. The "dead zone" of the T piece's ability to rotate is the same on either side of the board, in contrast to the I piece.
Unfortunately, it is not horizontally symmetrical.
The Square
The square is the only piece in the game that has true geometric symmetry, meaning a line can be drawn through it's centre at any angle, and one side will always reflect the other. It can also "spin" along it's central axis, although this does absolutely nothing as it fills the exact same gap no matter how it is rotated.
Along with the I piece however, it can not be placed on the exact centre of the board, which I find deeply displeasing as it is the only piece that comes close to having a complete set of symmetrical properties. The square is also an evil, soul destroying hell cube, the sole purpose of which is to rain down turds upon any semblance of order you manage to put together on the board.
In conclusion, I propose that despite appearances, there are no symmetrical pieces in Tetris.
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