A traditional way of proving the aperiodicity of a tiling is to show that a tiling can be "deflated" so that groups of smaller tiles are converted into a larger, single tile from the prototile set.
People then go on to say that "Since symmetries of a tiling apply also to its deflations, translational symmetries should apply to deflations of a tiling." So, if some translational symmetry works by moving every tile in some direction by some specific amount, they say that you can deflate the tiling until two of the points being translated fall within the same tile and since this isn't (trivially) a symmetry, the tiling is aperiodic. For example, they do it here, when proving the aperiodicity of an outstounding tiling.
Why is "symmetries of a tiling apply also to its deflations" true? For example, imagine a regular square lattice and I choose a translational symmetry of this (of which there are an infinite) where every square is moved to the one just right of it. But then, I deflate the tiling by repeatedly grouping four adjacent tiles into larger 2x2 squares until there are two points which are moved by the translation which stay within the same square. This would suggest, by the logic above, that the original translation was contradictory and thus that square lattices are aperiodic (they are clearly not).
What am I failing to understand?