An idealised stable uniaxial single-domain (SD) particle permits only two possible stable positions in which the magnetic moment can lie, either closely parallel or anti-parallel to the particle long (easy) axis. In real acicular SD particles, which have generally been regarded as uniaxial, this implicit two state feature has never been challenged, whilst there has been considerable debate concerning the mechanism of moment reversal between the two states. We present experimental results suggesting that acicular SD particles may actually have a range of several quantifiable stable (or metastable) orientations of the net magnetic moment. In order to help explain our experimental observations we present a new simple model of acicular SD particles, which gives quantitative predictions verified by further experiments. The model also appears to be relevant to other SD particle morphologies and crystal structures (such as hematite). A possible physical basis for our model in acicular particles may lie in non-uniform SD structures (such as the flower or vortex states). Small variations in the non-uniform SD structures available to a particle might allow a range of stable positions of the net moment. The results have several implications for rock magnetism and palaeomagnetism. Firstly, the new model can quantitatively account for several previously unexplained diverse phenomena exhibited by real acicular SD particles. These include the acquisition of gyroremanences and field-impressed anisotropy in dilute dispersions of such particles, as well as observations of transverse components of remanence in individual acicular SD particles. All these phenomena are theoretically impossible in idealised uniaxial SD particles. Interestingly, it appears that these phenomena could now be used to quantify the deviation of real acicular SD particles from ideal uniaxial behaviour and also, therefore, the deviation from a uniform SD structure. In hematite, observations of large field-impressed anisotropy appear to be quantitatively explained by the available positions of the moment in the basal plane. Secondly, computations of the ancient field vector and palaeointensity from remanence anisotropy techniques would not only be controlled by the shape and distribution of the particles, but also by the range of possible stable orientations of the net moment within each SD particle. Laboratory analogue remanences (and, we suspect, natural remanences) would be influenced by the range of possible stable moment positions. Quantifying the range of these stable moment positions, upon acquisition of laboratory or natural remanences, should lead to improved methods of computing the ancient field direction and palaeointensity in anisotropic rocks.