Deadlocks in SQL Server - how to avoid?
Due to the fact that both stored procedures issue 

UPDATE dbo.DeadlockTestMutex SET Toggle = 1 - Toggle WHERE ID = 1;


before attempting to execute any other code, the first stored procedure to successfully run that update will get an exclusive lock on dbo.DeadlockTestMutex and the other process will wait until that lock is released.  This allows each stored procedure to run to full completion without competition from the other stored procedure.