The task of obfuscating writing style using sequence models has previously been investigated under the framework of obfuscation-by-transfer, where the input text is explicitly rewritten in another style. These approaches also often lead to major alterations to the semantic content of the input. In this work, we propose obfuscation-by-invariance, and investigate to what extent models trained to be explicitly style-invariant preserve semantics. We evaluate our architectures on parallel and non-parallel corpora, and compare automatic and human evaluations on the obfuscated sentences. Our experiments show that style classifier performance can be reduced to chance level, whilst the automatic evaluation of the output is seemingly equal to models applying style-transfer. However, based on human evaluation we demonstrate a trade-off between the level of obfuscation and the observed quality of the output in terms of meaning preservation and grammaticality.