The symmetry of twins is described by chromatic point groups obtained from the intersection group {\cal H}^* of the oriented point groups of the individuals {\cal H}_i extended by the operations mapping different individuals. This article presents a revised list of twin point groups through the analysis of their groupoid structure, followed by the generalization to the case of allotwins. Allotwins of polytypes with the same type of point group can be described by a chromatic point group like twins. If the individuals are all differently oriented, the chromatic point group is obtained in the same way as in the case of twins; if they are mapped by symmetry operation of the individuals, the chromatic point group is neutral. If the same holds true for some but not all individuals, then the allotwin can be seen as composed of twinned regions described by a twin point group, that are then allotwinned and described by a colour identification group; the allotwin is then described by a chromatic group obtained as an extension of the former by the latter, and requires the use of extended symbols reminiscent of the extended Hermann–Mauguin symbols of space groups. In the case of allotwins of polytypes with different types of point groups, as well as incomplete (allo)twins, a chromatic point group does not reveal the full symmetry: the groupoid has to be specified instead.

中文翻译：

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双胞胎和异卵双胞胎的色对称性

孪生的对称性由通过映射不同个体的操作扩展的个体 {\cal H}_i 的定向点群的交集 {\cal H}^* 获得的色点群来描述。本文通过分析孪生点群的群形结构，提出了修订后的孪生点群列表，然后将其推广到 allotwins 的情况。具有相同类型点群的多型体的异体孪生可以用类似孪生的彩色点群来描述。如果个体的取向不同，则以与双胞胎相同的方式获得色点群；如果它们是通过个体的对称操作映射的，则色点群是中性的。如果对于某些但不是所有个体来说同样如此，则allotwin可以被视为由双点组描述的孪生区域组成，然后由颜色识别组分配和描述；然后，allotwin 由后者作为前者的扩展而获得的色群来描述，并且需要使用让人想起空间群的扩展赫尔曼-莫甘符号的扩展符号。在具有不同类型点群的多型体的异体孪生以及不完全（异体）孪生的情况下，彩色点群不会显示完全对称性：必须指定群形。