Andrew there are many aspects to your question and the answer will depend on the starting plant (species or hybrid etc). Firstly to start with orchids in general are not selfing and many species are obligate out-crossers i.e. selfed flowers will either not set fruit and or not produce any fertile seeds. This process is determined by genes belonging to the Self-incompatibility Locus (SI locus) to put it simply, the stigma has a mechanism to recognise the pollen from the same plant and it will either not allow the pollen to germinate or abort the flower to stop the process of fertilization.
In species where selfing and outcrossing both occurs, you can create homozygous plants for a certain characteristic by simply selfing them. For eg. When you self a plant the recessive allels which are not expressed owing to the presence of the dominant allel in the heterozygous parent (Dd) have a chance of being expressed in some of the offspring thus showing a different phenotype (dd).
When you consider hybrids things get very complicated and that is why you will see a lot of variation within the offspring of hybrids or from the selfings of the hybrids and different amounts of the parental genomes will be present and expressed in the siblings.
The stability and fertility of certain hybrids is dependent on the genetic similarity and closeness of the species hybridized. For eg. it is well known that the Asconopsis (Ascocentrum and Phalaenopsis hybrid) are poor plants for breedings, this is because Ascocentrum and Phalaenopsis are very distant related with different chromosome numbers and gene arrangement. While undergoing meiosis the cells are not able to divide the genetic material equally between the resulting gametes producing mostly non-functional gametes. They are better as seed parents than pollen parents as sometimes during gamete formation, meiosis is arrested and a 2n ovule remains as it is and is then fertilized by pollen giving offspring.