You can use a Mach-Zehnder interferometer with dissimilar number of reflections at each pathway. Therefore, your output will be a superposition of the form LG^{+1}+LG^{-1}, which is equivalent to a HG beam.
The input to the interferometer is only the LG{+1}. So, the first beam splitter of the interferometer creates two copies of the LG{+1}. One copy gets reflected an odd number of times and therefore it's OAM is reversed (it is now LG{-1}). The other copy gets reflected an even number of times and remains the same. Then, they are superimposed at the second beam splitter of the MZ interferometer creating the HG mode.
If you do not require an interferometric approach, you can use a mode converter using cylindrical lenses. See section 2.4 of Article Orbital angular momentum: Origins, behavior and applications
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