I guess there's no "to".
Sentence adopted and corrected.
I'd personally insert a comma.
FROM:
By mistake I boarded a train going in the opposite direction.
TO:
By mistake, I boarded a train going in the opposite direction.
@CK you are undoubtedly referring to rule # 3
"
Use a comma to set off introductory elements, as in "Running toward third base, he suddenly realized how stupid he looked."
It is permissible to omit the comma after a brief introductory element if the omission does not result in confusion or hesitancy in reading. If there is ever any doubt, use the comma, as it is always correct. If you would like some additional guidelines on using a comma after introductory elements, click HERE."
In other words, I can just as well leave it out, since it's correct as it stands.
http://grammar.ccc.commnet.edu/grammar/commas.htm
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