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palprime  and  zygodrome

First numbers that belong to both sets:

11, 1177711, 7722277, 7733377, 331111133, 772222277, 779999977, 992222299, 995555599, 11006660011, 11117771111, 11119991111, 11155555111, 11188888111, 11442224411, 11447774411, 11663336611, 11668886611, 11775557711, 11776667711, 11779997711, 11991119911, 11992229911, 33117771133, 33119991133, 33224442233, 33445554433, 33448884433, 33553335533, 33557775533, 77006660077, 77009990077, 77111111177, 77119991177, 77332223377, 77334443377, 77337773377, 77339993377, 77448884477, 77663336677, 77779997777, 77991119977, 99113331199, 99222222299, 99227772299, 99443334499, 99448884499, 99449994499, 99555555599, 99558885599, 99883338899, 99888888899, 1100011100011, 1100066600011, 1100999990011, 1111188811111, 1111666661111, 1112233322111, 1112255522111, 1112277722111, 1114411144111, 1118866688111, 1118888888111, 1122200022211, 1122777772211, 1133300033311, 1133344433311, 1144111114411, 1144466644411, 1155444445511, 1155500055511, 1155599955511, 1166688866611, 1177788877711, 1188811188811, 1199999999911, 3300222220033, 3300777770033, 3300888880033, 3331100011333, 3331177711333, 3332211122333, 3332255522333, 3334477744333, 3335555555333, 3335599955333, 3338800088333, 3338833388333, 3344666664433, 3355111115533, 3366777776633, 3388777778833, 3388999998833, 7700222220077, 7711444441177, 7711999991177, 7722111112277, 7722888882277, 7733300033377, 7733666663377, 7744466644477, 7744499944477, 7755500055577, 7755544455577, 7755599955577, 7766555556677, 7766633366677, 7771166611777, 7772255522777, 7774422244777, 7775522255777, 7775588855777, 7777111117777, 7777333337777, 7777788877777, 7777799977777, 7778844488777, 7778866688777, 7778888888777, 7788777778877, 7788844488877, 7788866688877, 7799222229977, 7799988899977, 7799999999977, 9922666662299, 9933777773399, 9955333335599, 9955666665599, 9966111116699, 9966888886699, 9988111118899, 9988333338899, 9991111111999, 9994433344999, 9997722277999, 9998811188999, 9998855588999, 110000888000011, 110009999900011, 110044222440011, 110044333440011, 110044444440011, 110066777660011, 110066999660011, 110099333990011, 110099444990011, 110099666990011, 111004444400111, 111008888800111, 111100111001111, 111100333001111, 111133444331111, 111133777331111, 111133888331111, 111166888661111, 111188111881111, 111188222881111, 111188888881111, 111199999991111, 111220000022111, 111337777733111, 111440000044111, 111779999977111, 111883333388111, 111991111199111, 112222111222211, 112222666222211, 112277111772211, 112277555772211, 113300111003311, 113300666003311, 113311888113311, 113330000033311, 113344222443311, 113377333773311, 113399888993311, 114400444004411, 114422111224411, 114422222224411, 114422333224411, 114433444334411, 114433777334411, 114455777554411, 114466555664411, 114466888664411, 114488666884411, 114488999884411, 114499111994411, 114499999994411, 115522444225511, 115522666225511, 115552222255511, 115555444555511, 115556666655511, 115577111775511, 116600111006611, 116600666006611, 116633888336611, 116633999336611, 116644666446611, 116660000066611, 116663333366611, 116666666666611, 116666888666611, 116677888776611, 116699999996611, 117700888007711, 117733111337711, 117733555337711, 117755111557711, 117755444557711, 117775555577711, 117779999977711, 117799111997711, 118811222118811, 118811444118811, 118811666118811, 118855333558811, 118855444558811, 118855555558811, 118855888558811, 118877444778811, 118877777778811, 118888666888811, 118888999888811, 119900111009911, 119911666119911, 119911888119911, 119911999119911, 119933666339911, 119944333449911, 119944999449911, 119966555669911, 119977333779911, 330011111110033, 330044666440033, 330055999550033, 330077111770033, 330077777770033, 330088666880033, 331111111111133, 331111555111133, 331114444411133, 331133333331133, 331133888331133, 331144111441133, 331144777441133, 331166666661133, 331166999661133, 331177111771133, 331199777991133, 332200333002233, 332222111222233, 332255222552233, 332255777552233, 332266333662233, 332266666662233, 332288888882233, 332299333992233, 333110000011333, 333322333223333, 333344444443333, 333355999553333, 333377222773333, 333377333773333, 333377888773333, 333388777883333, 333662222266333, 333668888866333, 333773333377333, 333889999988333, 334400333004433, 334411222114433, 334411777114433, 334411888114433, 334433666334433, 334444999444433, 334466333664433, 334466777664433.

Some larger examples :