Thermal rectification is an exotic thermal transport phenomenon which allows heat to transfer in one direction but block the other. We demonstrate an unusual dual-mode solid-state thermal rectification effect using a heterogeneous "irradiated-pristine" polyethylene nanofiber junction as a nanoscale thermal diode, in which heat flow can be rectified in both directions by changing the working temperature. For the nanofiber samples measured here, we observe a maximum thermal rectification factor as large as ~50%, which only requires a small temperature bias of <10 K. The tunable nanoscale thermal diodes with large rectification and narrow temperature bias open up new possibilities for developing advanced thermal management, energy conversion and, potentially thermophononic technologies.