Are genetic crosses between asexual organisms possible?
When it comes to viruses and bacteria, genetic cross is just as a vaguely defined concept as species. The latter is often defined on the percentage of the sequence similarity (for viruses) or the percentage of common genes (for bacteria).
If we speak about simply mixing genomes of different organisms, there are many possibilities, how this could happen. Just to give a couple of examples:

Some viruses integrate their genome into the host genome. In some cases these integrated genomes cease being functional viruses, and may evolve to be functional genes of the host. This is truly a genetic cross. See here, for example.
Bacteria and viruses routinely undergo horisontal gene transfer in which genes from one bacteria are integrated into the genome of the other one. A typical bacterial species has a few hundred core genes, and thousands of genes that are present only in some specimen or borrowed from other species, which significantly vary from one bacteria to another. Thus, every individual bacteria or bacterial community can be thought of as a genetic cross.