A Jinja would be more iconic. You see them far more often in modern photos and imagery - and promotionals - of Japan. Plus, they're tied to the legitimacy of the Chrsantanum Throne (a Japanese Emperor, to this day, must be Shinto as surely as the British Monarch must be Anglican, the three Scandinavian Monarchs Lutheran, the Dutch Monarch Reformed Dutch Christian, the Belgian and Spanish Monarchs, and Luxembourgish Grand Duke and Monacan and Liechtensteiner Prince Roman Catholic, the Moroccan, Jordanian, Saudi, Kuwaiti, Qatari, Bahraini, Bruneian, and all seven component Emirati and nine component Malaysian Monarchs Sunni Muslim, the Omani Monarch Obadhi Muslim, the Thai Monarch Theravada Buddhist, and the Tongan Monarch Wesleyan Christian. The King of Lesotho once boasted about being the only sitting, sovereign monarch free to choose his own religion).