Traditionally, cross-processing is a technique used by film photographers. A slide film, ordinarily developed with an E6 chemical process, is, instead, developed in chemicals meant for print film - C41. Sometimes it's the other way around. Often, the film is purposely over-exposed, or pushed a stop or two in the processing.
This usually results in high-contrast, deep saturation, and intriguing color shifts ranging from subtle to outrageous. It's all in the eye, and the mind, of the beholder. To paraphrase Mr. Dylan, "Something looks different here, but you don't know what it is. Do you, Mr. Jones."
While many flick film-shooting purists think it is irrelevant, some digital shooters are looking for post-processing techniques to simulate the effect. We at other's eyes believe in going for it. It's all about image making.