A dual clutch travel joint system includes an inner-mandrel and an outer-mandrel. The inner-mandrel includes a first connector on a downhole portion of the inner-mandrel and a second connector on an uphole portion of the inner-mandrel. The outer-mandrel includes a third connector that is positionable to mate with the first connector to apply torque to the inner-mandrel while the inner-mandrel is in a tension arrangement with the outer-mandrel within a wellbore. Additionally, the outer-mandrel includes a fourth connector positionable to mate with the second connector to apply torque to the inner-mandrel while the inner-mandrel is in a compression arrangement with the outer-mandrel within the wellbore.