Dipivaloylketene (2) is obtained by flash vacuum pyrolysis of furan-2,3-dione 6 and dimerizes to 1,3-dioxin-4-one 3, which is a stable but reactive ketene. The transannular addition and rearrangement of enols formed by the addition of nucleophiles to the ketene function in 3 generates axially chiral 2,6,9-trioxabicyclo[3.3.1]nonadienes (bisdioxines) 4. When arylamines are used as the nucleophiles under neutral conditions, decarboxylation occurs during the formation of bisdioxines 8. However, when water or alcohols are added to 3 under acidic conditions, bisdioxine-carboxylic acids and esters 10 and 11 are obtained. Acid hydrolysis of the bisdioxines proceeds through the addition of water to a C=C double bond and results in a second transannular oxa-Michael-type reaction and generation of tetraoxaadamantanes 5. This reaction is decarboxylative when free carboxylic acid functions are present in the bisdioxines, thus forming 21 and 22, but carboxylic acid derivatives are preserved to yield compounds 20, 23, 25, 28, and 29. A hydrogenolysis of the dibenzyl ester 23 yields the free dicarboxylic acid 24. The tetraoxaadamantanes are formed in high yields (65–95%) in most cases, but the addition of water to the concave inside of the bisdioxines becomes severely hindered in cyclic derivatives, so that the 38-membered ring compound 32 requires microwave heating at 170 °C to form tetraoxaadamantane 33, and the catenated compound 36 and calix[6]arene derivative 37 did not form tetraoxaadamantanes. The reaction mechanisms of bisdioxine and tetraoxaadamantane formation are discussed.