A roll-up covering for an architectural opening, comprising: at least a first outer elongate tape having a first end, a second end, a length that extends from the first end to the second end, a lateral width that extends perpendicular to its length, a thickness, and a first longitudinal axis extending along its length; at least a first inner elongate tape spaced from the first outer elongate tape, the first inner elongate tape having a first end, a second end, a length that extends from the first end to the second end, a lateral width that extends perpendicular to its length, a thickness, and a second longitudinal axis extending along its length; and a plurality of flexible slats attached to the first outer elongate tape and the first inner elongate tape, the slats each having a first end and a second end, an inner longitudinal edge and an outer longitudinal edge that extend from the first end to the second end and defining a length of the slat, an inner edge region along and in proximity to the inner longitudinal edge and an outer edge region along and in proximity to the outer longitudinal edge, and a width that extends from the inner longitudinal edge to the outer longitudinal edge, the slats oriented transversely with respect to the first inner elongate tape and first outer elongate tape when the roll-up covering is in an expanded configuration with the slat length extending in the same direction as the lateral width of the first outer elongate tape and first inner elongate tape, wherein the length of each slat is longer than the lateral width of at least one of the first inner elongate tape and the first outer elongate tape; characterized by: a plurality of elongate stiffeners, each stiffener having a length and a thickness; wherein each slat includes at least a first elongate stiffener attached to said slat within at least one of an inner edge region and an outer edge region of said slat, and said first elongate stiffener is positioned with its length extending in the same direction as at least one of the inner longitudinal edge and the outer longitudinal edge of respective said slat. at least a first outer elongate tape having a first end, a second end, a length that extends from the first end to the second end, a lateral width that extends perpendicular to its length, a thickness, and a first longitudinal axis extending along its length; at least a first inner elongate tape spaced from the first outer elongate tape, the first inner elongate tape having a first end, a second end, a length that extends from the first end to the second end, a lateral width that extends perpendicular to its length, a thickness, and a second longitudinal axis extending along its length; and a plurality of flexible slats attached to the first outer elongate tape and the first inner elongate tape, the slats each having a first end and a second end, an inner longitudinal edge and an outer longitudinal edge that extend from the first end to the second end and defining a length of the slat, an inner edge region along and in proximity to the inner longitudinal edge and an outer edge region along and in proximity to the outer longitudinal edge, and a width that extends from the inner longitudinal edge to the outer longitudinal edge, the slats oriented transversely with respect to the first inner elongate tape and first outer elongate tape when the roll-up covering is in an expanded configuration with the slat length extending in the same direction as the lateral width of the first outer elongate tape and first inner elongate tape, wherein the length of each slat is longer than the lateral width of at least one of the first inner elongate tape and the first outer elongate tape; characterized by: a plurality of elongate stiffeners, each stiffener having a length and a thickness; wherein each slat includes at least a first elongate stiffener attached to said slat within at least one of an inner edge region and an outer edge region of said slat, and said first elongate stiffener is positioned with its length extending in the same direction as at least one of the inner longitudinal edge and the outer longitudinal edge of respective said slat. a plurality of elongate stiffeners, each stiffener having a length and a thickness; wherein each slat includes at least a first elongate stiffener attached to said slat within at least one of an inner edge region and an outer edge region of said slat, and said first elongate stiffener is positioned with its length extending in the same direction as at least one of the inner longitudinal edge and the outer longitudinal edge of respective said slat.