We show that carbon-based nanostructured materials are a novel testbed for controlling thermoelectricity and have the potential to underpin the development of new cost-effective environmentally-friendly thermoelectric materials. In single-molecule junctions, it is known that transport resonances associated with the discrete molecular levels play a key role in the thermoelectric performance, but such resonances have not been exploited in carbon nanotubes (CNTs). Here we study junctions formed from tapered CNTs and demonstrate that such structures possess transport resonances near the Fermi level, whose energetic location can be varied by applying strain, resulting in an ability to tune the sign of their Seebeck coefficient. These results reveal that tapered CNTs form a new class of bi-thermoelectric materials, exhibiting both positive and negative thermopower. This ability to change the sign of the Seebeck coefficient allows the thermovoltage in carbon-based thermoelectric devices to be boosted by placing CNTs with alternating-sign Seebeck coefficients in tandem.