Whilst many techniques exist for generation of an optical vortex, there remains a need for new devices and methods that can also provide vortex generation withhigher powers, greater flexibility of wavelength, and generation beyondthe lowest-order Laguerre-Gaussian 𝐿𝐺01modeto address a broader range of practical applications.This work revealshow an all-mirror based interferometricmode transformation system can provide these propertiesincludingrevealing, for the first time,the generation ofa much richer set of vortex mode patterns than might have been thought possible previously.Anew developed theoreticalformulation, confirmed with excellent agreement by experimental demonstrationsin an imbalanced Sagnac interferometer,showsinterferometric transformation is possible for all orders of Laguerre-Gaussian 𝐿𝐺0𝑙modes into a rich set of high quality higher-order vortex and vortex superposition. The interferometric approachis shown to be configurabletoincrease or decrease vorticity. The new mathematical formulation provides the ability to perform a fullmodal power analysis of both the mode-transformed transmitted vortex and the complementary reflected beam at the Sagnac beamsplitter port.A discussion is made on the origin of the orbital angular momentum transferred to the vortex output from the Sagnac beamsplitter.