A rail fastening system for fastening a rail of a vertically movable cabin of an elevator system, the rail fastening system has a first clamp that can be connected to a holding structure, and it has a second clamp that can be connected to the rail, wherein the first clamp has a laterally extending first central part that is designed to be connectable to the holding structure, and the second clamp has a laterally extending second central part that is designed to be connectable to the rail, and both clamps each have a first and a second tab (12, 13, 22, 23) which are attached in a lateral direction at opposite ends of the relevant central part (11, 21), and the rail fastening system can be assembled in two alternative configurations, wherein, in the first configuration, the first tab of the first clamp is connected to the first tab of the second clamp at a first contact surface and the second tab of the first clamp is connected to the second tab of the second clamp at a second contact surface, and alternatively, in the second configuration, the first tab of the first clamp is connected to the second tab of the second clamp at a first contact surface and the second tab of the first clamp is connected to the first tab of the second clamp at a second contact surface, characterized in that a first plane which extends substantially in parallel through the first contact surface and a second plane which extends substantially in parallel through the second contact surface intersect at an intersection line, and in that the intersection line is perpendicular to a flat surface which extends substantially in parallel with the second central part. the rail fastening system has a first clamp that can be connected to a holding structure, and it has a second clamp that can be connected to the rail, wherein the first clamp has a laterally extending first central part that is designed to be connectable to the holding structure, and the second clamp has a laterally extending second central part that is designed to be connectable to the rail, and both clamps each have a first and a second tab (12, 13, 22, 23) which are attached in a lateral direction at opposite ends of the relevant central part (11, 21), and the rail fastening system can be assembled in two alternative configurations, wherein, in the first configuration, the first tab of the first clamp is connected to the first tab of the second clamp at a first contact surface and the second tab of the first clamp is connected to the second tab of the second clamp at a second contact surface, and alternatively, in the second configuration, the first tab of the first clamp is connected to the second tab of the second clamp at a first contact surface and the second tab of the first clamp is connected to the first tab of the second clamp at a second contact surface, characterized in that a first plane which extends substantially in parallel through the first contact surface and a second plane which extends substantially in parallel through the second contact surface intersect at an intersection line, and in that the intersection line is perpendicular to a flat surface which extends substantially in parallel with the second central part.