Finding Missing Inverse Relationships

Using EXCEPT could help you write this query.

In short, if any rows of the first select do not have matches with the inverse + the correct RelType & RelStatus , return them.

SELECT PartyId , OtherPartyId, RelType , RelStatus   
FROM dbo.Table
EXCEPT 
SELECT OtherPartyId , PartyId, RelType , RelStatus   
FROM dbo.Table;

DB<>Fiddle

Remember that if there are duplicate rows (E.G. 3 rows with an inverse relationship) these will not be returned.

Example


DDL

CREATE TABLE dbo.bla(ID INT IDENTITY(1,1) PRIMARY KEY NOT NULL,
                     PartyId int,
                     OtherPartyId int,
                     RelType int,
                     RelStatus int);

DML

INSERT INTO dbo.Bla(PartyId , OtherPartyId , RelType , RelStatus)
VALUES
(1111,2211           ,1       ,1),
(2211,1111           ,1       ,1),
(3344,4444           ,1       ,2),
(5555,2224           ,1       ,2),
(4444,3344           ,2      ,2),
(1111,2211           ,2      ,2);-- different type or status

Query

SELECT PartyId , OtherPartyId, RelType , RelStatus   
FROM dbo.Bla
EXCEPT 
SELECT OtherPartyId , PartyId, RelType , RelStatus   
FROM dbo.Bla;

Result

PartyId OtherPartyId    RelType RelStatus
1111    2211            2         2
3344    4444            1         2
4444    3344            2         2
5555    2224            1         2

Another solution could be using NOT EXISTS, E.G. When you also need the Id field.

SELECT Id,PartyId , OtherPartyId, RelType , RelStatus   
FROM dbo.Bla b
WHERE NOT EXISTS
(
SELECT * 
FROM dbo.Bla b2
where b.OtherPartyId = b2.PartyId
AND   b.PartyId = b2.OtherPartyId
AND b.RelType = b2.RelType
AND b.RelStatus = b2.RelStatus
);

DB<>Fiddle


The most common version of the query to do this would be something like:

INSERT INTO PartyRelationships (
    PartyId, 
    OtherPartyId, 
    RelType, 
    RelStatus
    )
SELECT pr.OtherPartyID, 
    pr.PartyId,
    pr.RelType,
    pr.RelStatus
FROM papr
LEFT JOIN PartyRelationships pri
    ON pr.PartyId = pri.OtherPartyId
    AND pr.otherPartyId = pri.PartyId
    AND pr.RelType = pri.RelType
    AND pr.RelStatus = pri.RelStatus
WHERE pri.partyId is null

The EXCEPT example by @randi-vertongen is also a method to do this, and may be more efficient. And one other method that is not an example of the above is the MERGE method

MERGE INTO PartyRelationships AS pr
USING PartyRelationships AS pri
    ON pr.PartyID = pri.OtherPartyId
    AND pr.OtherPartyId = pri.PartyId
    AND pr.RelType = pri.RelType
    AND pr.RelStatus = RelStatus
WHEN NOT MATCHED THEN
    INSERT(PartyId, OtherPartyId, RelType, RelStatus)
    VALUES(pri.OtherPartyId, pri.PartyId, pri.RelType, pri.RelStatus);

This is basically the same as the first example, just using the MERGE syntax. However, this operation is a self-merge of a sort, so this makes logical sense to me.

Whichever option you choose will work for what you have requested.