1. An image data processing device for converting original data as image data representative of an original image composed of a plurality of pixels that are arranged in a matrix, into processed data as image data representative of a processed image composed of a plurality of pixels that are arranged in a matrix, wherein the original data contains gradation-value data relating to optical properties of each of the pixels of the original image while the processed data contains multilevel-value data relating to properties of each of the pixels of the processed image, said image data processing device comprising: a processed-image gradation-value data generator operable to generate, based on the original data, gradation-value data relating to optical properties of each of the pixels of the processed image, by causing a plurality of pixel lines constituting the processed image to be sequentially subjected to a gradation-value-data-generation line processing, which is executed to generate the gradation value data of each of the pixels composing the subjected pixel line such that generation of the gradation-value data proceeds sequentially along the subjected pixel line in a gradation-value-data generation direction; a processed-image multilevel-value data generator operable to generate, based on the generated gradation-value data of each of the pixels of the processed image and according to an error diffusion technique, the multilevel value data relating to the properties of each of the pixels of the processed image, by causing the plurality of pixel lines constituting the processed image to be sequentially subjected to a multilevel-value-data-generation line processing, which is executed to generate the multilevel-value data of each of the pixels composing the subjected pixel line such that generation of the multilevel-value data proceeds sequentially along the subjected pixel line in a multilevel-value-data generation direction; and a line-processing direction controller operable to cause the gradation-value-data-generation and multilevel-value-data-generation line processings to be executed on each of the pixel lines in the same direction as the gradation-value-data and multilevel-value-data generation directions, and to invert each of the gradation-value-data and multilevel-value-data generation directions each time when the subjected pixel line is shifted from one to another in execution of a corresponding one of the gradation-value-data-generation and multilevel-value-data generation line processings.