You should be able to clean the fixed cup on the drive side without removing it. They tend to be installed very firmly hence the name fixed cup. If you friendly with a local shop they may loosen the cups for you.
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“One morning you wake up, the girl is gone, the bikes are gone, all that's left behind is a pair of old tires and a tube of tubular glue, all squeezed out"
Sugar "Kane" Kowalczyk