Tucked down a leafy road in Potomac, Maryland, the private museum Glenstone has barely a sign in the area to guide visitors, but it does have a beefy security guard at an imposing booth, from which the names of each hour’s reservations are relayed, one by one.

Up, with each confirmation, goes the gate at this 200-acre estate owned by Mitchell Rales, age 60, billionaire co-founder of a conglomerate called Danaher. Along with his wife Emily Rei Rales, 39, he has amassed some 800 works by the greatest names in modern and contemporary art: Willem de Kooning, Jackson Pollock, Mark Rothko, Alexander Calder, Henri Matisse, Brice Marden, Jasper Johns, Robert Rauschenberg, and more.

A curving drive leads past Split-Rocker, a huge sculpture by Jeff Koons of a head bedecked with 50,000 live flowering plants and an undulating work by Richard Serra. So private is Glenstone that it has no parking lot; we members of the 1 p.m. group leave our cars on the grass beside the French limestone–clad museum, which was designed by the late Charles Gwathmey. Across a lawn and a duck pond stands the Raleses’ capacious, Gwathmey-designed home, sleek and rather severe.

Eagerly we head into the gallery, to find empty walls. None of those modern masters, no paintings or sculptures at all. Only taut lengths of yarn from ceiling to floor, delineating trapezoids and rectangles.

This, in fact, is the work of the late minimalist Fred Sandback. It has filled the gallery’s seven rooms and 9,000 square feet of exhibition space for 15 months, while most of the Raleses’ $1 billion collection remains in storage. Before that the museum was closed for nine months.

Private museums get whopping tax breaks, which gives a taxpayer the right to ask: Is this a fair exchange? The U.S. Senate Finance Committee, under chairman Orrin Hatch, wondered the same thing. Last summer it concluded an investigation of 11 private museums, including Glenstone. The senator’s staffers seem more perplexed than when they started.