We present and discuss broadband infrared photometry in the 1-100 mum wavelength range, optical spectroscopy, and radio continuum observations of a sample of IRAS galaxies with unusual spectral energy distributions that peak near 60 mum. For inclusion in this sample of '60PKs', the galaxies must have satisfied the following criteria: IRAS flux ratios f60/f100 > 1 and 1 < f60/f25 < 4, and galactic latitude Absolute value of b > 10-degrees. In this paper, which is the first of a series, we show that the 60PKs are relatively scarce objects that represent about 2% of the spatial density of 60 mum-selected galaxies in the range L(60 mum) = 10(9) to 10(12) L., but have a far-infrared luminosity function of nearly identical shape. They are detected up to redshifts of 0.2. Besides having the usual high percentage of active galaxies (approximately 30% H II-region like, 50% Seyfert 2, 10% Seyfert 1, 10% unknown) associated with a flat 25 to 60 mum spectral index, the sample also includes most of the galaxies that have been found to have a dust-obscured broadline region. We show that the additional f60/f100 > 1 constraint selects galaxies with dust that is more centrally concentrated and exposed to a more intense radiation field than in most other IRAS galaxies. In particular, the cirrus component of the far-infrared radiation, which is typically the dominant contributor to the 100 mum emission from spiral galaxies, is negligible or missing. As discussed in Paper II [Heisler & Vader, 1993 (preprint)] of this series, this is consistent with the fact that the most distinctive optical signature of 60PKs is the absence of spiral structure: they tend to be peculiar and/or amorphous objects.