Yes, there was mutual respect on both sides. I read that Hitler issued a decree which expressly forbade soldiers from desecrating allied war graves not least because he had a soft spot for the soldiers of the Great War. Not that I think the average German infantryman was inclined to do so anyway. Most were actually religious, and many were superstitious. Desecrating graves was / still is a big cultural taboo.
(This is Hitler visting the Canadian War Memorial at Vimy Ridge. This was a PR stunt to disprove propaganda put out by the British that the Nazi’s were desecrating war memorials)
Having visited allied cemeteries in North East France, I recall the guides suggesting the German soldiers were actually very respectful. Most monuments and graves were completely untouched. Some did exhibit signs of war damage, though there’s no suggestion this was intentional.
There were isolated incidents of them destroying some memorials though. The Cross of Reconciliation at Lizerne is one example. The memorial recognised a chemical gas attack by the Germans in 1915 which killed French soldiers. The wording on the original memorial described the Germans as Barbarians which the occupying troops obviously took offence at. Another Australian war memorial at Mont Saint Quentin had an Aussie soldier bayonetting a German Eagle. The Nazis smashed the memorial .
The other most notable example was the Compiégne Wagon. The railway wagon in which the armistice was signed marking the end of WWI and German defeat in 1918. The wagon became an important national symbol for France and was moved to a museum. When the Germans invaded France in 1940, Hitler chose the exact same wagon in the exact same location as the place in which to sign the second armistice (you can imagine the mischievous grin on his face when he dreamed that one up). The wagon was moved to Germany, but was blown up with dynamite when the Allies began advancing into Germany.
 Adolf Hitler and Real History
 Monuments and Memorials on the Ypres Salient, Belgium
 Mont Saint-Quentin Australian war memorial – Wikipedia