The rare-earth tellurides are a family of low-dimensional compounds which host charge density wave (CDW) states. While CDW states in the rare-earth tri-tellurides (RTe₃) are relatively well-studied, less is known about CDW states in the rare-earth di-tellurides (RTe₂). The RTe₂ compounds are quasi two dimensional materials with a crystal structure which consists of alternating insulating rare-earth block layers and single–layer conducting Te sheets. Here we present our scanning tunneling microscopy measurements on CeTe₂. Our measurements detect two unidirectional, perpendicular, spatially separated CDW states which smoothly evolve from one to the other. Our measurements shed light on the connection between CDW states and local strain.