Shortly before he died, John Pezzullo wrote an Excel spreadsheet that calculates the ratio and confidence interval (and p value) for a comparison of either 1 two ratios and their confidence intervals or 2 two means and their confidence intervals.
The Excel spreadsheet (with one tab for means and another for ratios) is here.
With an explanation here.
For the example in the question, using the Ratios tab in the Excel spreadsheet for 0.52 (95% CI 0.25 to 1.08) compared to 0.68 (95% CI 0.52 to 0.89) we get 0.76 (0.35-1.67) for A/B and 1.31 (0.60-2.85) for B/A, with p = 0.50.
