## 1 Introduction  
Periodic structures have been extensively used to engineer the desired electromagnetic response in a wide range of devices; for instance, filters [1], frequency selective surfaces [2], antennas [3], absorbers [4], and lenses [5]. However, the usefulness of periodic structures is often limited by their inherent frequency dispersion, as well as their relatively low effective refractive index. Recently, it was demonstrated that the introduction of higher symmetries into a periodic structure can alleviate these limitations. A periodic structure is said to possess a higher-symmetric if it is invariant after a translation and an additional geometrical operation [6; 7; 8; 9]. Consequently, the translational unit cell of a higher-symmetric structure can be divided into \(\xi\) sub-unit cells, where \(\xi\) is the (entire) order of the symmetry. The translational period of the higher-symmetric structure is then given by \(p=\delta\xi\), where \(\delta\) is the size (period) of the sub-unit cell. A generalized Floquet theorem was presented in [9], stating that the Bloch modes in a higher-symmetric structure are eigenmodes to both the translation operator of the unit cell, \(\mathcal{T}_{p}\), and the compound operator applied to the sub-unit cell, \(\mathcal{C}_{\delta}\). The most relevant result of this theorem is that it suffices to analyze the sub-unit cell of a higher-symmetric structure, which can significantly reduce the required computational resources. A direct consequence of the existence of \(\xi\) sub-unit cells within a unit cell is the absence of band gaps between the first \(\xi\) modes of the structure [10]. Mainly, two types of higher symmetries have been investigated for electromagnetic purposes: glide- and twist-symmetries. The additional geometrical operation in a glide and twist-symmetric structure is a mirroring in a plane and a rotation around an axis, respectively. A glide symmetry is inherently of order \(\xi=2\), while a twist symmetry can have higher orders, \(\xi\geq 2\), as happens for the four-fold twist-symmetric structure illustrated in Figure 1.  
Recent studies of glide-symmetric structures have demonstrated the absence of the stop-band between the first and second modes [11; 12]. Additionally, it has been demonstrated that glide-symmetric structures are less dispersive [12; 13] and can provide a higher effective refractive index [13; 14; 15; 16], effective permeability [17], and level of anisotropy [18], compared to their non-glide counterparts. These properties have been used to design planar lenses in [13; 19; 20; 21; 22]. Furthermore, while the stop-band between the first two modes is suppressed in glide-symmetric structures, a large stop-band exists at higher frequency [23; 24; 25]. This stop-band has been used to design low-cost gap waveguides [26] and filters [27] and to reduce the leakage between waveguide flanges [28; 29]. These attractive properties have inspired the development of several semi-analytical methods for the analysis of glide-symmetric structures using circuit models [25; 30; 31], mode-matching [32; 33; 34; 35], or the multimode transfer-matrix approach [36; 37; 38]. These methods provide a fast means of studying glide-symmetric structures and give insight into the physics of glide symmetry. Similar properties have been demonstrated in twist-symmetric structures. However, since the order of the twist symmetry can be changed, an increased control of the electromagnetic response is enabled, compared to translational and glide-symmetric structures. For instance, the absence of stop-bands between the first \(\xi\) modes (\(\xi\geq 2\)) was demonstrated in [39; 40; 41]. Additionally, reduced frequency dispersion and an increased effective refractive index in twist-symmetric structures were demonstrated in [41]. These properties have been used to design twist-symmetric filters [40], phase-shifters [42], miniaturized helix antennas [43], lenses [44], and polarization filtering optical fibers [45].