Elaborating the legal position of the Writ jurisdiction of the Supreme Court and the High Courts pointed out by our KK!HR, I would like to add that it can be exercised even against a private company when it is charged with or delegated to do certain public functions is the ratio decidendi of the judgment of the Supreme Court in N.C.Mehta v Union of India and Others.
However, in industrial disputes, the writ jurisdiction of High Court is invoked under Article 226 by either the employer or the employee only when either one is aggrieved by the orders of the authorities under the IDA,1947. Position is the same in respect of other labor laws too where the decision of an authority is said to be final.