So, while the two steps outlined are the core steps to doing bi-directional blast queries, I want to point out that this is not "bidirectional by default". Bi-directional by default would imply that when you run a single blast query against a database, it would do the reverse query of the database against the query FASTA by default. This is not the case.
It may be clearer to specify that what is being done is more commonly referred to (at least back in the day when I did this routinely) as reciprocal BLAST.
What is the point of doing this exercise? Are you planning to transfer annotations from one strain to others?
If you are looking for genome-wide similarities/rearrangements then using mauve is perfect solution.