Based on your example data, here's one way to do it:
grid.col <- setNames(rainbow(length(unlist(dimnames(mat)))), union(rownames(mat), colnames(mat)))
par(mar = c(0, 0, 0, 0), mfrow = c(1, 2))
# original image
chordDiagram(mat, grid.col = grid.col)
# now, the image with rotated labels
chordDiagram(mat, annotationTrack = "grid", preAllocateTracks = 1, grid.col = grid.col)
circos.trackPlotRegion(track.index = 1, panel.fun = function(x, y) {
xlim = get.cell.meta.data("xlim")
ylim = get.cell.meta.data("ylim")
sector.name = get.cell.meta.data("sector.index")
circos.text(mean(xlim), ylim[1] + .1, sector.name, facing = "clockwise", niceFacing = TRUE, adj = c(0, 0.5))
circos.axis(h = "top", labels.cex = 0.5, major.tick.percentage = 0.2, sector.index = sector.name, track.index = 2)
}, bg.border = NA)
Result:
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