Batch alter Relationship Contact Permission only
After some debugging it has turned that  we have exactly same situation described here How to change the direction of a relationship type with existing data , but we weren't aware that changing Relationship direction via GUI doesn't make change in a database. Sounds like another civicrm bug.  

We've ended up by direct Mysql intervention with this query (in case someone will face this issue in future): 

UPDATE 
 civicrm_relationship 
SET 
  contact_id_a=contact_id_a+contact_id_b,
  contact_id_b=contact_id_a-contact_id_b,
  contact_id_a=contact_id_a-contact_id_b
WHERE ** ... conditions set ...  **