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Rotation symmetric Boolean functions (RSBFs) that are invariant under circular translation of indices have been used as components of different cryptosystems. In this paper, even-variable-balanced RSBFs with maximum algebraic immunity (AI) are investigated. At first, we give an original construction of 2m-variable-balanced RSBFs with maximum AI. Then we improve the construction to obtain more 2m-variable-balanced RSBFs with maximum AI, and these new RSBFs have higher non-linearity than all previously obtained RSBFs. Further, we generalise our construction of 2m-variable RSBFs to a new construction that can generate any even-variable RSBFs.