The aetiology of cancer involves intricate cellular and molecular mechanisms that apparently emerge on the short timescale of a single lifetime. Some of these traits are remarkable not only for their complexity, but also because it is hard to conceive selection pressures that would favour their evolution within the local competitive microenvironment of the tumour. Examples include 'niche construction' (re-programming of tumour-specific target sites) to create permissive conditions for distant metastases; long-range feedback loops of tumour growth; and remarkably 'plastic' phenotypes (e.g. density-dependent dispersal) associated with metastatic cancer. These traits, which we term 'paradoxical tumour traits', facilitate the long-range spread or long-term persistence of the tumours, but offer no apparent benefit, and might even incur costs in the competition of clones within the tumour. We discuss three possible scenarios for the origin of these characters: somatic selection driven by specific selection regimes; non-adaptive emergence due to inherent vulnerabilities in the organism; and manipulation by putative transmissible agents that contribute to and benefit from these traits. Our work highlights a lack of understanding of some aspects of tumour development, and offers alternative hypotheses that might guide further research.