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We find in the case of W (110) previously overlooked anomalous surface states having their spin locked at right angle to their momentum using spin-resolved momentum microscopy. In addition to the well known Dirac-like surface state with Rashba spin texture near the-point, we observe a tilted Dirac cone with circularly shaped cross section and a Dirac crossing at 0.28× within the projected bulk band gap of tungsten. This state has eye-catching similarities to the spin-locked surface state of a topological insulator. The experiments are fortified by a one-step photoemission calculation in its density-matrix formulation.