How to Optimise Query
The optimizer can consider many plan alternatives (including ones with multiple seeks) but for disjunctions (OR predicates) it does not consider plans involving index intersections by default.  Given the indexes:

CREATE CLUSTERED INDEX cx 
ON dbo.DispatchLink (DispatchLink1, ContractLink1);

CREATE NONCLUSTERED INDEX nc1 
ON dbo.DispatchLink (DispatchLink2, ContractLink2);


We can force index seeks (assuming SQL Server 2008 or later):

SELECT * 
FROM dbo.Dispatch AS d
INNER JOIN dbo.DispatchLink AS dl WITH (FORCESEEK) ON 
    (d.DispatchId = dl.DispatchLink1 AND d.ContractId = dl.ContractLink1)
    OR (d.DispatchId = dl.DispatchLink2 AND d.ContractId = dl.ContractLink2);




Using your sample data, the seek plan costs at 0.0332551 units compared with 0.0068057 for the scan plan:



There are all sorts of possible query rewrites and hints we can try.  One example of a rewrite to promote an option the optimizer does not consider for the original plan is:

SELECT * 
FROM dbo.Dispatch AS d
CROSS APPLY
(
    SELECT TOP (1) * FROM
    (
        SELECT * FROM dbo.DispatchLink AS dl
        WHERE dl.DispatchLink1 = d.DispatchId
        AND dl.ContractLink1 = d.ContractId
        UNION ALL
        SELECT * FROM dbo.DispatchLink AS dl
        WHERE dl.DispatchLink2 = d.DispatchId
        AND dl.ContractLink2 = d.ContractId
    ) SQ1
) AS F1;


This execution plan does not seek the second index if it finds a match on the first:



This may perform very slightly better than the default FORCESEEK plan.

Without adding any new indexes, we can also force a seek into the Dispatch table:

SELECT * 
FROM dbo.DispatchLink AS dl
JOIN dbo.Dispatch AS d WITH (FORCESEEK) ON
    (d.DispatchId = dl.DispatchLink1 AND d.ContractId = dl.ContractLink1)
    OR (d.DispatchId = dl.DispatchLink2 AND d.ContractId = dl.ContractLink2);




This may be better or worse than the first example depending on things like how many rows are in each of the tables.  The APPLY + TOP improvement is still possible:

SELECT * 
FROM dbo.DispatchLink AS dl
CROSS APPLY
(
    SELECT TOP (1) * FROM
    (
        SELECT * FROM dbo.Dispatch AS d
        WHERE dl.DispatchLink1 = d.DispatchId
        AND dl.ContractLink1 = d.ContractId
        UNION ALL
        SELECT * FROM dbo.Dispatch AS d
        WHERE dl.DispatchLink2 = d.DispatchId
        AND dl.ContractLink2 = d.ContractId
    ) SQ1
) AS F1;