A dual-tip catheter equipped with both a rotating cutting bit and a cavity for a probing guidewire tip for passage through chronic total occlusions or other refractory atherosclerotic plaque from diseased arteries. The catheter's guidewire tip and rotating cutting bit reside safely within an outer protective catheter sheath head when not in use, and alternately either the cutting bit or the guidewire tip will protrude out of the same distal opening of the sheath head as needed. By first drilling through refractory occlusions, and then inserting a guidewire tip into the occlusion opening formed by the cutting bit, the catheter can force the guidewire tip past the occlusion. After the guidewire tip has been fully inserted, the dual-tip catheter can be withdrawn, and an alternate catheter inserted and directed up the guidewire to the occlusion site, where the alternate catheter can perform further occlusion enlargement or therapeutic activity.