The effect of genetic operators other than selection, such as mutation and recombination, on the genotype-phenotype map is considered. In particular, when the genotypic fitness landscape exhibits a ``symmetry'', i.e. many genotypes corresponding to the same phenotype have equal fitness values, it is shown that such operators can break this symmetry. The consequences of this ``induced symmetry breaking'' are investigated. Specifically, it is shown that it generically leads to an increase in order or self-organization in the system and to the phenomenon of orthogenesis. Additionally, it is shown that it potentially leads to a more robust evolution circumventing some of the problems of brittleness. The above points are supported by explicit, analytic results associated with some simple one and two-locus models and also by some much more complicated numerical simulations.