Is there a way to make this select retrieve the same results with a single seek?

I'd try this but I have no idea if it will be more efficient. You need the DISTINCT to remove duplicates, so the UNION ALL might be more appropriate, no need for two distinct operations:

SELECT DISTINCT 
    JobRunId = @JobRunID, 
    d.*
FROM dbo.TestUpload
  CROSS APPLY 
    (   SELECT 
          ThingAName AS Name, 
          ThingAType AS [Type], 
          ThingAGranularity AS Granularity
      UNION                            -- or UNION ALL
        SELECT 
          ThingBName, 
          ThingBType, 
          ThingBGranularity
    ) AS d 
WHERE JobRunId = @JobRunID ;

UNION ALL plan:

UNION plan:

Use cross apply to unpivot columns into rows

SELECT --DISTINCT most probably
  JobRunId,
  ut.Name, 
  ut.[Type], 
  ut.Granularity
FROM dbo.TestUpload
CROSS APPLY (
  SELECT ThingBName AS Name, 
  ThingBType AS [Type], 
  ThingBGranularity AS Granularity
  UNION ALL 
  SELECT ThingAName AS Name, 
  ThingAType AS [Type], 
  ThingAGranularity AS Granularity
  ) ut
WHERE JobRunId = @JobRunID