It's hard to think of the rose we know under this name as a climber, but some of the early refs do refer to it as a climbing rose.
The most mature plants I know tend to grow quite low and wide - throwing out moderately long, strong, rigid canes horizontally. Perhaps this length of cane might be interpreted as a climbing cane but it's not a typical climber by any stretch.
Many who grow it find it a bit awkward and out of place among more typical Tea roses.
Some who grow the rose cut the long sideways growth back regularly to keep it within bounds (and reduce the tripping hazard!), but a lot of bloom is sacrificed in the process.
The bloom is lovely though - and the perfume's sweet.
Here's a description from the 1901 Dingee & Conard catalogue (have added to refs) where it is referred to as a climbing rose:
"A remarkable new climbing rose - Madam Wagram or Climbing Paul Neyron, No 162 (Bernaix). - This is perhaps the most extraordinary Rose introduced in many years. The originator places it among Tea Roses, although in his description he himself says that it is one of the largest Tea Roses with Hybrid Perpetual blood. Indeed it is a very large Rose, nobly formed and well filled with firm, broad petals. The bud is round and thick and stands erect on a firm stem. The flower reminds one of that great Rose Merveille de Lyon, so large is it, in fact, it is almost as large as Paul Neyron; opens freely and endures well on the stem when cut. Color fresh satiny rose passing into fleshy and Chinese red, with a silvery sheen over-spreading the whole; edge of petals appears white. It is a strong grower and more after the the type of the Hybrid Tea and Hybrid Perpetual than a Tea Rose. It is a grand rose for all general purposes and, like many others, must be seen before its real value will be recognised by hesitating lovers of the Rose. It will, however, attain success. The genuine stock is very scarce. 25 cts. each; 3 for 60cts; two-year-old plants, 50 cts. each."