Symmetrical Scales are called like this because they are built out of structures that repeat, for instance the two most common ones are the whole tone scale (consisting only – as the name implies – out of whole tone steps) and the octatonic scale (consisting of half-whole or whole half step building blocks). As a consequence these scales can only be transposed a limited amount of times. For instance, there are only two different whole tone scales. All other transpositions of that scale will generate the same scale (just with different starting points).
One of my favourite and a little lesser known of these scales is the 1-3-scale and 3-1-scale. The later one e.g. allows you to create quite alot of fantastic sounding polytonal situations as you have for example Cm, Em and Abm triads in this scale. The resulting compositions or sound worlds when using these scales is always “mystical”, “outerworldly”, “ambiguous” etc. as these symmetrical structures also have the consequence of these scales not implying a clear root note. They always stay tonally ambiguous and “open”.
To gather experience with working with these scales and also develop a feeling of their sound, it helps to simply improvise over these pitches, try to find chords that are inherent in that scale and build blocks out of this. Also, thinning out the scales and which tones you use might make it easier to get started. These scales work to colourize individual chords, sections or even whole pieces on this. For instance, check out Debussy’s Voiles to get a feeling for a piece that is based for a long time on the whole tone scale.
If you want to dive deeper into this topic and explore its worlds composer Oliver Messiaen based quite a lot of his work on such scales, which he called Modes of limited transposition.