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You could also use Transpose and Summarize to deal with the duplicates:
ID each record, transpose so Field1 and Field 2 are in the same column, sort so your values are in the same order for each record, concatenate so each record is back on one row, grab a unique and use text to columns to separate the fields back out. Do a select at the end to drop unwanted columns and rename back to the original.
Note that this might flip-flop what's in Field1 vs Field2. From your data it didn't look like this was important, but if it is then keep the RecordID in the Select tool and join it back to the data right after the Record ID tool.
@danrh - Thanks, Dan. Reordering the pair ID's by adding a record ID, transposing the data, sorting the data by the record ID and pair ID's, then finally finding unique pair ID's is what helped me get what I needed.