There is in fact a straightforward invariant that works in this circumstance - the 'classic' invariant for tiling problems. Checkerboard-color the grid, and note that all other pieces (the $\color{cyan}{\text{I}}$, $\color{green}{\text{S}}$s, $\color{blue}{\text{J}}$s, and the square) take up an equal number of black and white squares in all orientations; by contrast, the $\color{purple}{\text{T}}$ always takes up three squares of one color and one square of the other. This implies that for any number of rows (each of which have an equal number of squares of each color) to be perfectly filled, an even number of $\color{purple}{\text{T}}$ pieces must be used.

There is in fact a straightforward invariant that works in this circumstance - the 'classic' invariant for tiling problems. Checkerboard-color the grid, and note that all other pieces (the $\color{cyan}{\text{I}}$, $\color{green}{\text{S}}$s, $\color{blue}{\text{J}}$s, and the square) take up an equal number of black and white squares in all orientations; by contrast, the $\color{purple}{\text{T}}$ always takes up three squares of one color and one square of the other. This implies that for any number of rows (each of which have an equal number of squares of each color) to be perfectly filled, an even number of $\color{purple}{\text{T}}$ pieces must be used.

EDIT: As pointed out by Gilad, this approach only solves the 'static' issue of tiling a rectangle with tetrominoes including the $\color{purple}{\text{T}}$, not the 'dynamic' region-filling that Tetris provides for; see his excellent solution for why this matters if intermediate rows can be cleared and partial-pieces left behind.

There is in fact a straightforward invariant that works in this circumstance - the 'classic' invariant for tiling problems. Checkerboard-color the grid, and note that all other pieces (the I, Ss, Ls, and the square) all take up an equal number of black and white squares in all orientations; by contrast, the T always takes up three squares of one color and one square of the other. This implies that for any number of rows (each of which have an equal number of squares of each color) to be perfectly filled, an even number of T pieces must be used.

There is in fact a straightforward invariant that works in this circumstance - the 'classic' invariant for tiling problems. Checkerboard-color the grid, and note that all other pieces (the $\color{cyan}{\text{I}}$, $\color{green}{\text{S}}$s, $\color{blue}{\text{J}}$s, and the square) take up an equal number of black and white squares in all orientations; by contrast, the $\color{purple}{\text{T}}$ always takes up three squares of one color and one square of the other. This implies that for any number of rows (each of which have an equal number of squares of each color) to be perfectly filled, an even number of $\color{purple}{\text{T}}$ pieces must be used.